Strongly coupled dense matter and hedgehog black holes S. Prem Kumar (Swansea U.) February 1, 2011, DAMTP, Cambridge (work in progress with P. Benincasa) S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Introduction The behaviour of cold dense baryonic matter is one of the outstanding issues in theoretical physics. Equation of state for bulk nuclear matter (e.g. Stephanov ’07; Sch¨ afer ’05 ) T QGP 0.17 GeV 77 Quark Nuclear CFL Matter matter 0.9 GeV μ Passing from nucleons to quarks requires non-perturbative treatment. Standard lattice techniques suffer from infamous sign problem. [Alternate approaches: Imaginary µ ; stochastic quantization..] S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Another approach: theories without a sign problem, e.g. QCD with isospin chemical potential; theories with real matter respresentations. Models within the framework of the AdS/CFT correspondence Require large- N (colours) and large- N f (flavours): Veneziano limit: N f → ∞ and N → ∞ with N f N fixed. + + ........ 2-2 g b ~ ∑ c (N / N) N f b,g Not obvious if weakly coupled string dual exists when N f N ∼ 1. S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Outline • Review N = 4 theory with N = 2 matter (D3-D7 system). • Reducing the flavour group by “smearing”. • Phase structure at weak coupling. • Strong coupling picture at finite baryon density. • Outlook S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Theory with fundamental matter: D3-D7 system x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 × × × × D3 × × × × × × × × D7 D3-D7 open strings: N f hypermultiplets i = 1 , 2 . . . N f , ( Q i , ˜ Q i ) → ( N , ¯ N ) of SU ( N ). (Karch-Katz ’02) N = 4 theory coupled to N = 2 matter N f � √ √ Q i Φ 3 Q i + m ˜ Q i Q i � � 2 ˜ W = + 2 Tr (Φ 3 [Φ 1 , Φ 2 ]) . i =1 Flavour symmetry: U ( N f ) ≃ U (1) B × SU ( N f ) R-symmetry: SU (2) R × U (1) R × SU (2) Φ S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
N = 1 “smeared” theory: D-brane picture (see N´ u˜ nez-Paredes-Ramallo 2003 . . . , 2010 review) • Six directions transverse to D3-brane: z 1 = x 4 + ix 5 , z 2 = x 6 + ix 7 , z 3 = x 8 + ix 9 z 3 z 2 z 1 N D7 's f m N D3 's Smearing U(1)N U(N ) f f SU (3) • Obtained by action of U (1) × SU (2) on the orientation vector (0 , 0 , 1). S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
SU (3) U (1) × SU (2) ≃ CP 2 . As N f → ∞ the smearing “orbit” New superpotential Yukawa couplings : N f 3 � Q i Φ a Q i → X † · � � � i ˜ d � X ˜ Q X � λ a Φ Q X a =1 i =1 X = Ω (1 , 0 , 0) T and Ω ∈ SU (3). with � Theory has SU (3) × U (1) global symmetry; and U (1) N f whose diagonal combination is baryon number For N f ∼ N c , perturbative β -function has Landau pole, β λ ∼ N f N λ 2 . We must treat the theory with a UV cut-off. S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Some weak-coupling intuition Perturbative study of gauge theories on S 3 × R can provide some intuition for what to expect. Studying large- N theories on finite volume is natural from the point-of-view of AdS/CFT correspondence. Most famously, Hawking-Page transition in AdS 5 has been connected to Hagedorn/deconfinement transition of free gauge theory on S 3 × S 1 β . (Aharony-Marsano-Minwalla-Papadodimas-Van Raamsdonk ’03) What is the thermodynamics of weakly coupled large- N theory with N f ∼ N flavor fields? S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Lightest field on S 3 × S 1 β is the Wilson/Polyakov loop: � U = exp i A 0 ≡ ( α 1 , α 2 . . . α N ) β Integrating out KK harmonics and matter fields results in a unitary matrix model � � � � d ℓ Tr ln(1 − U e − β ( ǫ ℓ − µ ) ) Z [ U ] = dU exp − N f ℓ + d ℓ Tr ln(1 − U † e − β ( ǫ ℓ + µ ) ) + . . . �� When N f , N → ∞ , complex saddle point configuration dominates the integral: { α i } lie on a continuous contour in the complex plane. (Hands-Hollowood-Myers ’10) S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Fixed (low) T and varying µ { α i } distributed on a contour C - density function ρ ( z ) with pole(s): < Tr U > =0 < Tr U > =0 Pole Pole Low μ μ increasing confined phase Gross-Witten "deconfinement" transition When µ → mass of lightest scalar mode, Bose-Einstein condensation occurs, and occupation number → ∞ . <Tr U >=1 • Theory makes transition to Higgs phase (akin to moving from “Coulomb to Higgs branch”) z=1 Pole S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Phase diagram T deconfinement of N=4 3rd order GW Deconfined Higgs phase Confined (runaway ?) μ The deconfined phase goes all the way to T = 0. Taken seriously, this suggests a dual black hole state at low temperatures. The Higgs phase potential seems to be runaway at tree level. This could be corrected radiatively. S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
The “smeared D3/D7” system at strong coupling Start with the probe picture of D7-branes in AdS 5 × S 5 , wrapping S 3 ⊂ S 5 . For massive flavours, the “slipping angle” θ ↔ fermion bilinear ˜ ψ i ψ i , smoothly caps off Mass � 2.56 Θ 1.5 S 3 θ 1.0 0.5 5 S z 0.1 0.2 0.3 0.4 0.5 Writing S 5 as a U (1)-fibration over CP 2 d Ω 2 5 = ds 2 CP 2 + ( d ψ + A C P 2 ) 2 SU (3) × U (1) manifest; broken to SU (2) Φ × SU (2) R × U (1) R . S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
SU (3) × U (1) is preserved by the smeared, backreacted solutions of N f D7’s obtained from d 10 x √− g 10 | ˜ �� � � C 8 ∧ ˜ S = S IIB − T D7 Ω 2 | + Ω 2 , ˜ Ω 2 is the “smearing form” controlling the D7-distribution dF 1 = − g s Ω 2 F 1 = N f p ( r )( d ψ + A C P 2 ) ds 2 = c 1 dr 2 + c 2 ds 2 1 , 3 + c 3 ds 2 C P 2 + c 4 ( d ψ + dA C P 2 ) 2 , F 5 = 16 π N α ′ 2 (1 + ∗ )Ω 5 and dilaton exhibits UV Landau pole. With massive flavours for some r < r crit ( m ), the geometry is AdS 5 × S 5 , with constant dilaton and F 1 = 0 (no D7-branes). S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
(global) AdS Backreacted D7 geometry What happens when chemical potential µ � = 0 ? For probes, µ � = 0 corresponds to a radial electric field F 0 r in the DBI action. (Mateos-Myers-Thomson ’06-’07; Karch-O’bannon ’06-’07) For large enough quark density, the electric field induces F1-spike on the D7-brane S D7 → nS F1 [ Nambu − Goto ], Mass : 3.34 Mass : 3.34 Θ field 1.5 4 3 1.0 2 0.5 1 z z 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
• In global AdS this poses a potential problem due to Gauss’s law; need baryon vertices to absorb the string flux: We need to describe the combined D7-F1-D5(baryon vertex), a potentially complicated configuration. Backreacted D7 geometry • Two crucial simplifications: ⇒ smearing of strings. Smearing of D7-branes = IIB equations automatically include flux sourced by D5-branes at the origin. S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
Smeared F1’s and IR geometry • Expect IR geometry to be sourced by a backreaction of strings only (no D 7 ′ s and F 1 = 0) • First approximation: consistent SO (6)-symmetric smearing ansatz, �� d 10 x √− g | ˜ � S IIB − nNN f � B 2 ∧ ˜ S = Ω 8 | + Ω 8 2 πα ′ ˜ Ω 8 = Ω 3 ∧ Ω 5 The SO (6) will be actually be broken by matching conditions with the UV flavour-brane background. The D7-brane physics is frozen/decoupled in this limit. Equivalent to looking for gravity dual of a state with O ( N 2 ) static quarks in N = 4 SYM. S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
The consistent IIB background With strings uniformly smeared on compact transverse space, how is Gauss’s law for B 2 satisfied: � C-S term ∼ C 4 ∧ F 3 ∧ H 3 Equation of motion for B 2 allows H 3 = B 2 = 0, provided nN f N 1 2 πα ′ Ω 8 = F 5 ∧ F 3 , 32 π G 10 So, F 3 = # n N f Ω 3 ↔ nN f D5-branes/baryons. Therefore, we are looking at a high density state, energy density ∼ O ( N 2 ), containing O ( N ) baryons. S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
SO (6)-symmetric ansatz for metric (Einstein frame) ds 2 = − g tt dt 2 + g rr dr 2 + e 2 σ d Ω 2 3 + e 2 η d Ω 2 5 Action for metric and dilaton, S = N 2 � dr dt √ g rr g tt e 3 σ +5 η � (3 σ ′ + 5 η ′ ) 2 R (2) + g rr � 4 − 3 σ ′ 2 − 5 η ′ 2 − 1 � + 6 e − 2 σ + 20 e − 2 η − 8 e − 10 η 2 φ ′ 2 − Q 2 e φ e − 6 σ − 2 Q e φ/ 2 e − 3 σ − 5 η � √ Q ≡ n 2 λ N f π N Four equations and one constraint and we look for smooth solutions. S. Prem Kumar (Swansea U.) Strongly coupled dense matter and hedgehog black holes
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