HUGS 2012 Dualities and QCD Josh Erlich LECTURE 5
Outline The meaning of duality in physics (Example: The Ising model) Quark-Hadron duality (experimental and theoretical evidence) Electric-Magnetic Duality (monopole condensation and confinement) The AdS/CFT correspondence (gauge/gravity duality, holographic QCD)
Building a Bottom-Up Model Step 1: Choose 5D gauge group and geometry. Tower of vector mesons are identified with tower of Kaluza-Klein gauge bosons. SU(2) isospin 5D SU(2) gauge theory Conformal in the UV Anti-de Sitter space near its boundary Can choose geometry by matching spectrum to Pade approx of SU(2) current-current correlator in deep Euclidean regime q 2 m 2 ρ. Result: geometry = slice of AdS space (Shifman; JE,Kribs,Low; Falkowski,Perez-Victoria).
Building a Bottom-Up Model To include the full chiral symmetry, not just the vector subgroup, SU(2) SU(2) chiral symmetry SU(2) SU(2) 5D gauge group Additional tower of gauge bosons tower of axial-vector mesons. (5D parity 4D parity) (Also describes pions after symmetry breaking)
Building a Bottom-Up Model Step 2: Include pattern of chiral symmetry breaking Hint from AdS/CFT: 4D operator 5D field q i q j Scalar fields X ij, bifundamental under SU(2) SU(2) Background profile for X ij : Non-normalizable mode source L 4D m ij q i q j Normalizable mode VEV q i q j (Klebanov,Witten; Balasubramanian et al.) The scalar field background explicitly and spontaneously breaks the chiral symmetry.
Building a Bottom-Up Model For definiteness, we need to choose 5D mass of scalar field. AdS/CFT: m 2 X = qq( qq 4) in units of AdS curvature. In the UV, qq = 3, so we choose m 2 X = 3. Note: This choice is made for definiteness, but is not necessary.
Building a Bottom-Up Model In summary, the model is: SU(2) SU(2) gauge theory in slice of AdS 5 with background bifundamental scalar field. S = d 5 x g 1 2g 2 5 Tr (L MN L MN + R MN R MN )+Tr( D M X 2 3 X 2 ) ds 2 = 1 z 2 dxµ dx µ dz 2, < z < z IR Model parameters: X 0 (x, z) = m q 2 z + qq 2 z3 g 5, m q, qq, z IR (JE,Katz,Son,Stephanov; DaRold,Pomarol)
Building a Bottom-Up Model Vector: V a µ = La µ + R a µ 2 Axial-Vector: A a µ = La µ R a µ Pions: Mixture of longitudinal part 2 of A a µ and scalar field X
Soft-Wall AdS/QCD In the Hard Wall model m 2 n n 2 To obtain a linear Regge trajectory, the geometry can be modified while coupling to a dilaton background. (Karch,Katz,Son,Stephanov 06) S = d 5 x ge Φ(x,z) L Φ 0 (z) z 2, g MN = AdS 5 Metric Low-energy predictions are similar to hard-wall model
Hard-Wall (5D tree level) With strange quark mass parameter Abdidin and Carlson 09
Hard/Soft-Wall (5D tree level) Pion Form Factor 1 0.8 0.6 F! (Q 2 ) 0.4 0.2 0 0 1 2 3 4 5 Q 2 (GeV 2 ) from Kwee and Lebed, arxiv:0807.4565 Solid black and blue curves: Hard wall model Dotted red and green curves: Soft wall model See also Grigoryan,Radyushkin 08
Hard/Soft-Wall (5D tree level) 4 3.5 3 2!b" + 0, hard 2! b" c 0, hard 2!b" + 0, soft 2!b" c 0, soft Gravitational Form Factors and Generalized Parton Distributions 2!b" 0 (fm -1 ) 2.5 2 1.5 1 0.5 From Abidin and Carlson, arxiv:0801.3839 0 2.5 2 0 0.2 0.4 0.6 0.8 1 b(fm) 2!b" + 1, hard 2!b" c 1, hard 2!b" + 1, soft 2!b" c 1, soft Top: p + and charge densities of Helicity-0 rho mesons in hard and soft wall models 2!b" 1 (fm -1 ) 1.5 1 Bottom: Same for Helicity-1 rho mesons 0.5 0 0 0.2 0.4 0.6 0.8 1 b(fm) See also Lyubovitskij, Vega,...
Hard-Wall (5D tree level) Can determine meson radii from behavior of form factors near q 2 = 0. Hard wall model: r 2 π charge =0.33 fm 2 r 2 π grav =0.13 fm 2 r 2 ρ charge =0.53 fm 2 r 2 ρ grav =0.21 fm 2 r 2 a 1 charge =0.39 fm 2 r 2 a 1 grav =0.15 fm 2 H. Grigoryan and A. Radyushkin; Z. Abidin and C. Carlson 07, 08
Universality in AdS/QCD? Some observables are truly universal, i.e. independent of details of model. Famous Example: Viscosity to Entropy Density η/s Finite temperature spacetime horizon Prediction, independent of details of spacetime geometry: Kovtun, Son and Starinets 04 η s = 1 4π Another example: Electrical Conductivity to Charge Susceptibility σ/χ Kovtun and Ritz 08 σ χ = 1 4πT d d 2
Universality? Observables are roughly independent of X mass MeV 1400 1200 1000 800 600 400 200 m a 1 F F 1/2! 1/2 a 1 5 4 3 2 1 g!"" 2.7 2.8 2.9 3.0 3.1 3.2 3.3 # 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 # From JE,Westenberger 09 m 2 X = ( 4). = 3 is the value set by matching to the UV.
Light-Front AdS/QCD Brodsky and De Teramond have discovered some remarkable similarities between wave equations in AdS and the light-front wave equation for multiparton states. Example: Two-parton state b Longitudinal momentum fraction x = k + /P + Define light-front kinematical variable ζ x(1 x)b 2 Wavefunction: ψ(x, ζ, ϕ) =e imϕ X(x) φ(ζ)/ 2πζ
Light-Front AdS/QCD Light-front wave equation P µ P µ ψ = M 2 ψ d2 dζ 2 1 4L2 4ζ 2 + U(ζ) φ(ζ) =M 2 φ(ζ) Confining interaction potential Scalar wave equation in soft-wall model: 2 z3 e ϕ(z) z e ϕ(z) z 3 z + µr z ψ(z) =M 2 ψ(z) Identifying these equations gives the AdS coordinate z a kinematical interpretation
Phases of QCD According to Wikipedia Baryon chemical potential
More of the QCD Phase Diagram T <u! 5 d>=0 A < " >=0 <u! 5 d>=0 m " µ I FIG. 1. Phase diagram of QCD at finite isospin density. Son,Stephanov 00 Positive fermion determinant allows for lattice study Alford,Kapustin,Wilczek; Kogut,Sinclair; de Forcrand, Stephanov,Wenger; Detmold et al.
Systems with Isospin Neutron Stars (Low temp, Large isospin) Quark-Gluon Plasma at RHIC,LHC (Higher temp, Smaller isospin)
QCD w/ Isospin Chemical Potential Small (Son&Stephanov, 2000) T =0: : Can use Chiral Lagrangian [ ] Chemical potential couples to isospin number density. µ I,T J (3) 0 = ψγ 0 τ 3 ψ L 4D = f π 2 4 Tr ( ν Σ ν Σ ) + m2 π f π 2 4 Σ =exp(2iπ a τ a /f π ) Tr ( Σ + Σ ) 0 Σ = 0 Σ µ I 2 (τ 3Σ Στ 3 ).
QCD w/ Isospin Chemical Potential Son-Stephanov ansatz: Σ =cosα + i(τ 1 cos φ + τ 2 sin φ)sinα Results: Σ = 0if µ I >m π Phase transition is second order c 2 s > 1/3 speed of sound
Isospin Chemical Potential in the S = Hard Wall Model d 5 x g Tr Scalar field background { DX 2 +3 X 2 1 4g 2 5 ( ) } F 2 L + FR 2 X 0 (z) = 1 2 ( mq z + σz 3) 1 2 v, Vector combination of gauge fields V a M =1/2(La M + Ra M ) Gauge choice Linearized equation of Motion Vector field background Source for J (3) 0 ( ) L a z = Rz a = 0 ( ) 1 z z zvµ a ( V 3 0 (z) =c 1 + c 2 2 z2 1 z α α Vµ a =0 c 1 = µ
Isospin Chemical Potential in the Hard Wall Model Can calculate all the properties of the isospin condensate phase discussed earlier; results agree with other approaches. (Albrecht,JE)
Other Observations Baryons are solitons in extra dimensions - like Skyrmions (Piljin Yi s talk;sakai,sugimoto; Nawa et al.; Pomarol, Wulzer) HoloQCD may address qualitative questions like chiral symmetry restoration (D.K.Hong et al.; Shifman,Vainshtein; Klempt) Can add higher dimension 5D operators or modify geometry to match power corrections in Operator Product Expansions (Hirn,Sanz)
Dualities Lecture 5 Summary QCD Towers of bound states identified by quantum numbers, mass Hidden local symmetry: Vector mesons act like massive gauge bosons (Sakurai; Bando et al.) Chiral symmetry breaking has implications at low energies Holographic QCD Towers of Kaluza-Klein modes identified by quantum numbers, mass Vector mesons modeled by Kaluza- Klein modes of gauge fields (Polchinski,Strassler;Son,Stephanov; Brodsky,De Teramond; etc.) Chiral symmetry breaking has implications at low energies Turkey Weinberg sum rules Tofu Weinberg sum rules
Dualities Lectures Final Summary Dualities are alternative descriptions of the same physical systems. Sometimes one description is easier to analyze than another (e.g. strong-weak dualities) Dualities also motivate new models of strongly interacting physics, as in holographic QCD.