Holography with backreacted flavor

Transcription

1 Holography with backreacted flavor Diana Vaman MCTP, University of Michigan Miami 2006 Conference p. 1

2 Based on: Holographic Duals of Flavored N=1 super Yang-Mills: Beyond the Probe Approximation JHEP 0502:022,2005, hep-th/ , B.Burrington, J.T.Liu, L.Pando Zayas and D.V. Regge Trajectories for Mesons in the Holographic Dual of Large-N c QCD JHEP 0506:046,2005, hep-th/ , M. Kruczenski, L.Pando Zayas, J.Sonnenschein and D.V. The D3/D7 Background and Flavor Dependence of Regge Trajectories Phys.Rev.D72:026007,2005, hep-th/ , I.Kirsch and D.V. Holograpic phase transition with backreacted flavor J. Shao and D.V., hep-th/ p. 2

3 Motivation: The AdS/CFT is most conspicuously a duality without open strings. p. 3

4 Motivation: The AdS/CFT is most conspicuously a duality without open strings. As a consequence the dual gauge theory has no fields in the fundamental representation of the gauge group. p. 3

5 Motivation: The AdS/CFT is most conspicuously a duality without open strings. As a consequence the dual gauge theory has no fields in the fundamental representation of the gauge group. Where are the quarks? Adding flavor dof: To reintroduce the open strings, one adds probe branes to the AdS/CFT scenario [KK]. But, with probe branes, we are forced to consider only cases N f N c. We re missing: -chiral phase transition; -quantum moduli of susy gauge theories with flavor dof... Can one do better? p. 3

6 Motivation: The AdS/CFT is most conspicuously a duality without open strings. As a consequence the dual gauge theory has no fields in the fundamental representation of the gauge group. Where are the quarks? Adding flavor dof: To reintroduce the open strings, one adds probe branes to the AdS/CFT scenario [KK]. But, with probe branes, we are forced to consider only cases N f N c. We re missing: -chiral phase transition; -quantum moduli of susy gauge theories with flavor dof... Can one do better? Yes, include the backreaction of the probe branes. Backreated flavor branes dynamical (virtual) light quarks. p. 3

7 Outline The D3-D7 system at T = 0 (beyond the probe approximation): susy variations a Monge-Ampere eqn the warp factor: an analytic solution Regge trajectories The D3-D7 system at T 0 (beyond the probe approximation): the solution the quark condensate chiral phase transitions p. 4

8 The D3-D7 system (at T=0) Idea: modify the AdS background by including the supergravity fields that are sourced by the probe branes. p. 5

9 The D3-D7 system (at T=0) Idea: modify the AdS background by including the supergravity fields that are sourced by the probe branes. Any Dp-brane is charged under a certain supergravity field which is a p + 1 form. p. 5

10 The D3-D7 system (at T=0) Idea: modify the AdS background by including the supergravity fields that are sourced by the probe branes. Any Dp-brane is charged under a certain supergravity field which is a p + 1 form. D7 branes source a cplx scalar field: the axion-dilaton τ = χ + ie φ. A purely N f D7 brane geometry: ds 2 = dx 2 + e φ dzd z τ(z) = i N ( ) f z 2π ln, ρ L = e 2π/(g sn f ), z = ρe iϕ. ρ L Pathology at ρ = ρ L. For N f 12, solution with a well-behaved dilaton j(τ) = (z/ρ L ) N f [GSVY: stringy cosmic strings]. p. 5

11 The fully localized D3-D7 system: D3 D7 The ansatz: 6 ds 2 = h 1/2 (x m )dx µ dx µ + h 1/2 (x m ) g mn dx m dx n, m,n=1 (F 5 ) M1...M 5 = ǫ M1...M 5 M 6 F M 6 + 5ǫ [M1...M 4 F M5 ], τ = τ(x m ) p. 6

12 The fully localized D3-D7 system: The ansatz: D3 D7 ds 2 = h 1/2 (x m )dx µ dx µ + h 1/2 (x m ) 6 m,n=1 g mn dx m dx n, (F 5 ) M1...M 5 = ǫ M1...M 5 M 6 F M 6 + 5ǫ [M1...M 4 F M5 ], τ = τ(x m ) The program: solve the Killing spinor equations; the space transverse to the D3 s is Kähler; the problem factorizes: first solve for the 6d Kähler potential (MA eqn), then solve for the warp factor. p. 6

13 In detail: the susy variations: P m (1 Γ m )ǫ = 0, ( s µ ǫ 8 n log(h) + 1 ) 2 F n (Γ µ Γ n )ǫ = 0, (6) m ǫ s ( 1 2 F mǫ + 8 n log(h) + s ) 2 F n (1 Γ n m )ǫ i 2 Q mǫ = 0, where P M = i M τ 2 τ 2, Q M = Mτ 1 2τ 2 and γ 5 ǫ = sǫ. = s = 1, F m m h, ǫ = cov. const. The F 5 ansatz is the same as without the D7 branes. The 6d space to D3 is Kähler. = The problem factorizes. p. 7

14 In detail (cont.): use the integrability condition of the 6d susy variation ( ) R m n = P m P n m n ln(detg (6) ) = m n ln Iτ which yields a Monge-Ampere eqn for the Kähler potential det( m n K) = f(z) f( z)i(τ) Info about the D7 branes configuration determines the rhs. solve for the warp factor from the Einstein equation N c (6) h = δ 6 (x m x m 0 ) detg(6) p. 8

15 The method for solving the D3-D7 system: -choose a holomorphic axion-dilaton with appropriate monodromies -solve the Monge-Ampere equation -solve the warp factor. p. 9

16 The method for solving the D3-D7 system: -choose a holomorphic axion-dilaton with appropriate monodromies -solve the Monge-Ampere equation -solve the warp factor. Example: A single stack of D7 branes -the source I(τ) = 1 g s N f 4π ln(z 3 z 3 ) -the Kähler potential: K = z 1 z 1 + z 2 z 2 + f(z 3 z 3 ) obeys 3 3f = 1 g s N f 4π ln(z 3 z 3 ) -the solution ds 2 (6) = dz 1d z 1 + dz 2 d z 2 + ( 1 N ) f g s 4π ln(z 3 z 3 ) dz 3 d z 3 p. 9

17 The warp factor Solve the transverse Laplacean (6) = e Ψ(z 3 z 3 ) 3 3, e Ψ = 1 g s N f 4π ln(z 3 z 3 ) Have the D3 and D7 placed at the origin: SO(4) SO(2) symmetry: r 2 = z z 2 2, z 3 = ρe iϕ. Solve the Green s function: (6) G(ρ, ϕ, r; ρ, ϕ, r ) = N c g(6) δ 4 ( r r )δ(ρ ρ )δ(ϕ ϕ ) by decomposing into spherical harmonics G = 1 + Q D3 d 4 qe i q( r r ) e il(ϕ ϕ ) y l, q (ρ, ρ ) l where ( 2 ρ 2 1 ρ ρ ) l2 + V (ρ) + ρ 2 y l,q (ρ; ρ ) = 0, V (ρ) = ( 1 N ) f g s 2π log ρ q 2. p. 10

18 With a change of variable x = log(ρ/ρ L ), the warp factor reduces to solving the diff eqn [GP]: 2 xy q (x) = λxe 2x y q (x), λ = N f 2π ρ2 Lq 2, where we defined y(x) y l=0, q=0 (ρ(x); ρ = 0). There are 2 asymptotic soln: for x : y c, y ax + b, which can be found in terms of a unif. convergent series on x (, 0]: y(x) = 2π 2 λ n e 2nx p n (x), n=0 where the polynomials p n (x) are defined recursively by ( 4n 2 + 4n d ) dx + d2 dx 2 p n (x) = xp n 1 (x), n = 1, 2, 3,..., p 0 (x) = x x 0 γ. p. 11

19 In the near-core region (ρ ρ L x ) with ρ 2 = λxe 2x, the warp factor is given by y = c 1 I 0 ( ρ) + c 2 K 0 ( ρ) h(r, ρ) = 1 + Q D3 = Q D3 (r 2 + ρ 2 e Ψ ) 2 dqq 2J 1(qr) K 0 ( λxe 2r 2x ) ρ y( ) ρ p. 12

20 Decoupling limit The N = 4 fields correspond to 3-3 strings. In addition we have 3-7 strings and 7-7 strings. In the limit α 0, only the 3-3 strings and 3-7 (N = 2 hypermultiplets) strings localized at the D3D7 intersection remain: g Dp = g s ls p 3 the D7 worldvolume gauge theory decouples. Also g s << 1, g s N c =fixed. The SO(6) R-symmetry group is broken to SU(2) hyper SU(2) R U(1) R. The beta-function (λ = gy 2 M N c) is non-vanishing: βn=2 λ N c β N=2 = 1 ( ) 2 λ N f. 2π 4π N c For N f /N c fixed, the field theory is neither conformal nor asymptotically free. the existence of a UV Landau pole (Λ L ): λ(λ) = 1 N f ln(λ L /Λ) p. 13

21 Matching pathologies Compare with the wv action for a probe D3 placed in the D3/D7 supergravity background: S D3 = T D3 d 4 σe Φ det(g ab + F ab ) + T D3 C 4 + C 0 F F [ T D3 d 4 σ(2πα ) 2 1 ] 4 e Φ F ab F ab + χf ab F ab +..., where χ = N f 2π ϕ, e Φ = N f 4π log ρ2 L ρ 2. The D3-brane action relates g 2 Y M = 4πe Φ, θ Y M = 2πχ Upon identifying also Λ 2 = ρ 2 /(2πα 2 ), Λ L = ρ 2 L /(2πα 2 ), the running of the gauge coupling follows from the logarithmic behavior of the dilaton. The U(1) R chiral anomaly is reflected by the non-trivial axion profile. p. 14

22 Mesons Mesons of low spin: fluctuations of probe D7. Mesons of large spin: open spinning strings ending on probe branes. A phenomenological model of the meson: a spinning open string with massive end-points. ω For a string derivation of this model, introduce probe branes in the confining supergravity background of choice. ω D brane r 0 p. 15

23 Regge trajectories Solve the classical string eom and find a U-shaped spinning string configuration. The corresponding Regge trajectories J(E 2 ) get a correction due to the quark masses ( vertical arms ) E = 2T g ω J = 2T g ω 2 ( arcsinx + x 1 ) 1 x 2 ( arcsinx + 32 x 1 x 2 ), x = speed of the endpoints : the mass-loaded Chew-Frautschi formula. Reggeregimex 1 : J = 1 ( 2 E πt g π ( mq E ) 3/2 + π 1 π ) m Q E +... where m Q = (1 x 2 )T g /(ωx). p. 16

24 Flavor dependence of Regge trajectories Consider an open string rotating in the near-horizon limit of the D3/D7 background, ending on a D7 probe at ρ R from the stack of D7. the four-dimensional spacetime: dx µ dx µ = dt 2 +dr 2 +R 2 dϕ 2 +dz 2 ; the field theory the set-up: N f massless flavors plus an additional massive flavor whose mass is proportional to ρ R ; assume a large spin for the meson (semiclassical approx validity); ansatz for a string rotating with constant angular velocity ω is t = τ, ϕ = ωτ, R = R(σ), r = r(σ), ρ = ρ(σ), with world sheet coordinates σ and τ classical Nambu-Goto action: L = T s (det g) = Ts (1 ω 2 R 2 )(h 1 R 2 + r 2 + (1 N f 2π log ρ)ρ 2 ). p. 17

25 the energy and the angular momentum of the spinning string ( E = dσ ω L ) ω L = dσ ω h E 1 R (σ) 2 + r (σ) 2 + e ψ ρ (σ) 2, J = dσ L ω = dσ R 2 h E 1 R (σ) 2 + r (σ) 2 + e ψ ρ (σ) 2, with E = 1 R 2, T s = 1 and h( r, ρ) = Q D3 ( r 2 + ρ 2 e Ψ( ρ) ) 2. eom in the gauge R = σ: ( ) d E 2 d R L R r = E2 2L r h 1, ( ) d E 2 ( d R L eψ R ρ = E2 2L ρ h 1 + ( R ρ) 2 ρ e Ψ). = r 0 is a solution of the equation of motion. p. 18

26 the remaining eom coordinates z = 1/ ρ, we find d d R ( E 2 L eψ R z 1 ) = E2 2L z2 ( z h 1 + ( R z 1 ) 2 z e Ψ). Neumann bc for the directions to the probe D7, Dirichlet bc for the directions = the string end to D7. solve the string profile z( R) substitute into the energy and angular momentum = obtain E = E( J) fix the quark mass: m = ρ R ε g00 g ρρ dρ = ρ R ( 1 ε g s N f 2π log ρ)1/2 dρ p. 19

27 E/m 2.5 Q J/ 1/4 λ Figure 1: Chew-Frautschi plot for N f = 0, 1, 2, 3, 5, 10 additional massless flavors. The straight line represents the N f = 0 trajectory for small spin values. All graphs approach the horizontal line E = 2m. p. 20

28 Tension Nf Figure 2: String tension in dependence of the number of flavors N f. p. 21

29 Finite temperature localized D3D7 system The non-extremal D3-D7 solution in the Einstein frame is given by ds 2 10 = h(r) 1/2 ( f(r) dt 2 + d xd x) + h(r) 1/2» f(r) 1 dr 2 + R 2 sin 2 β dω R2 dβ 2 «+ R 2 cos 2 β 1 2α + (5 4ln(R cos β))α 2 dφ 2 ) + O(α 3 ) h(r) = 1 + L4 R 4 + α2 2L4 ln(r) R 4 + O(α 3 ) f(r) = 1 R4 0 R 4 α2 2R4 0 ln(r) R 4 + O(α 3 ) e Φ = 1 2αln(R cos β) + 2α 2 (1 ln(r cos β)) + O(α 3 ) χ = 2αφ + O(α 3 ) C (4) = Q 1 + L4 R 4 α2 2L4 ln(r) R 4 «1 dt dx 1 dx 2 dx 3 + O(α 3 ), Q 2 = L4 + R 4 0 L 4 α = g sn f 2π = λn f N c p. 22

30 The probe brane embedding The probe brane Lagrangian S = T D7 Z dt d 3 x dσ ω 3 e Φq det g αβ + T 7 Z = T D7 Z C (8) dt d 3 x dr ω 3 R 3 sin 3 β e Φp «1 + R 2 fβ 2 (e Φ 1 2α 2 )(sinβ + R cos ββ ) R cos(beta) R sin(beta) 0th order in alpha 1st order 2nd order 2 p. 23

31 The quark condensate Far away, the profile of the probe D7 is given by z m + c r 2, r The composite mass quark is equal to the energy of a string stretched between the D3 stack and the D7 probe, far away from the black hole horizon ( m). The quark condensate is equal to the vev of the hyperquark bilinear ψψ ψψ = δe δm q Parametrizing the probe brane embedding as z = z(r), we have ( L[z(r)] = T D7 ω 3 r 3 e Φ(z) 1 (r + z dz r2 + z 2 dr )2 ) + f (z r dz ) dr )2 1 p. 24

32 δe = δz T D7 r 3 e Φ z(r + zdz/dr) fr(z rdz/dr) (z) r2 + z 2 (r + zdz/dr) 2 + f(z rdz/dr) 2ω 3 = δm q (2πl 2 s)2ct D7 e Φ/2 (m)ω 3 where we have used that δz = δm = 2πl 2 s e Φ/2 δm q. Then, the quark condensate is given by ψψ = (2πl 2 s) 2cT D7 e Φ/2 (m)ω 3 The effect of the backreacted flavor branes on the quark condensate: Consider a Minkowski brane lying far outside the black hole horizon: r= r=0 z(r) = z(r = ) + δz(r) = m + δz(r), δz(r) 1, m R 0 z(r) m + c 0 + c 1 α + c 2 α 2 r 2, r «c 0 = m3 96 ǫ2, c 1 = m3 ǫ 576 (72 3ǫ), c 2 = m3 ǫ 144(1 + ln(m)) 10ǫ(1 + 3ln(m)) 576 The quark condensate value decreases at each order in α. p. 25

33 Chiral phase transitions at finite temperature c m c 1 m Blue curve: Minkowski branes (have a vanishing S 3 ) Red curve: black branes (have a vanishing S 1 ) The phase transition remains of first order: the quark condensate jumps discontinuously accros the transition. p. 26