Isospin chemical potential Isospin chemical potential in - PowerPoint PPT Presentation

Isospin chemical potential Isospin chemical potential in holographic “QCD” in holographic “QCD” Marija Zamaklar University of Durham based on work with Ofer Aharony (Weizmann) Cobi Sonnenschein (Tel Aviv) Kasper Peeters (Utrecht) 0709.3948 and in progress Galileo Galilei Institute, May 6th 2008

Introduction AdS/CFT integrability “high precision tests” N = 4 a perfect “purest” generator of a huge # of integrable struct. non-AdS/non-CFT (direct) applications to realistic gauge theories zero temperature and chemical potentials ( T = 0 , µ = 0 ) glueball spectra Csaki et al. masses of hadrons (mesons) Karch & Katz... hadron form factors Polchinski & Strassler finite-temperature and µ = 0 theories Son, Starinets, . . . (viscosity of quark-gluon plasma) THIS TALK ! Finite chemical potential

Setup: (I) Pure Glue pure QCD — i.e. no matter do not know geometry   instead, consider 4+1 dim max. susy YM in IR, reduces to     compactify on circle pure QCD, scalars     impose anti-periodic bdy. cond. for fermions and fermions decouple dual to near-horizon geometry of non-extremal D4-brane, doubly Wick rotated

The geometry [Witten, Sakai & Sugimoto, . . . ] � 3 / 2 � d u 2 � u � R � � 3 / 2 � d s 2 = η µν d X µ d X ν + f ( u )d θ 2 � f ( u ) + u 2 dΩ 4 + R u “ u Λ ” 3 world-volume f ( u ) = 1 − u: radial direction u our 3+1 world bounded from θ is a compact below u ≥ u Λ Kaluza-Klein circle

Several remarks  R 3 D 4 = πg s l 3 R D 4 : s N c   Solution characterised  � R 3 � 1 / 2 R =2 π → M Λ = 2 π by two parameters: D 4  R :   3 u Λ R Relation to gauge-theory parameters: non-extremality of D-brane: angle θ identified with size of S 1 on D4 (i.e. M KK ) set by R period R to avoid conical singularity YM N c = R 3 λ ≡ g 2 D 4 α ′ R Regime of validity: sugra OK if R 2 ≡ R 3 D 4 /R ≫ α ′ λ ≫ 1 (max curvature at the wall) valid as long as e φ = g s ( u/R D 4 ) 3 / 4 < 1 (min coupling at the wall) Problem : M KK ∼ M glueball ∼ M meson ∼ M Λ ∼ 1 /R cannot decouple KK modes !

Overview u = energy scale θ u u = u Λ focus here IR “wall” � � 3 / 2 Λ = R 3 / 2 D4 u 1 / 2 R D4 u 2 u Λ Λ

Setup : (II) Introducing matter–Sakai-Sugimoto model Add D8 flavour (probe) branes to D4 stack strings between flavour & colour branes in fund. rep. of flavour & colour group Solve for the shape of the D8 D 4 : 0 1 2 3 θ − − − − − D 8 : 0 1 2 3 − 5 6 7 8 9 � �� � � �� � flat curve u ( θ ) coord. sys. adapted to D8 θ, Ω 4 θ D8 D8 Solution to the 1st order equation gives embedding u ( θ ) wall direction

Symmetry encoded in geometry Asymptotically exhibits full chiral symmetry SU ( N f ) L × SU ( N f ) R Bending of the brane encodes spontaneous symmetry breaking in gauge theory in a geometrical way SU ( N f ) L SU ( N f ) isospin spectrum of fluctuations contains ( π ± , π 0 ) Goldstone bosons SU ( N f ) R Brane geometry also reproduces chiral symmetry restoration above T > T c SU ( N f ) L SU ( N f ) R

Low spin mesons Spectrum is known only in the limits: Low-spin mesons: fluctuations on and of the flavour brane Fluctuations governed by Dirac-Born-Infeld action of the flavour brane � d 5 x e − φ � − det ( g µν + 2 πα ′ F µν ) + S Wess-Zumino S = V S 4 � d 4 x d z √− g F µν F ρλ g µρ g νλ + . . . = V S 4 Expand world-volume fields in modes meson spectrum & action

Effective action for light mesons Decompose the gauge fields � G ( n ) F µν = µν ( x ) ψ ( n ) ( u ) , n � B ( n ) F uµ = µ ( x ) ∂ u ψ ( n ) ( u ) , n Fourier transform & factor out polarisation vectors, � � � �� u − 1 / 2 γ 1 / 2 ( ω 2 − � d 4 k ˜ ˜ B ( m ) B ( n ) k 2 ) ψ ( n ) − ∂ u u 5 / 2 γ − 1 / 2 ∂ u ψ ( n ) = 0 . µ µ � �� � a Sturm-Liouville problem mass spectrum of mesons

High spin mesons Spectrum is known only in the limit: Sigma model ( semiclass ) high-spin glueballs ( closed ) & mesons ( open ) q ¯ q meson: u f 1 m q ∼ u f 1 − u Λ region I u f 2 “projected” u Λ region II √ N.B. High spin mass M high ∼ λM Λ vs. low spin mass M low ∼ M Λ ∼ M KK

Part II: Part II: Turning on an isospin chemical potential Turning on an isospin chemical potential Chiral Langrangian Chiral Langrangian

Isospin vs Baryon chemical potential Why isospin chemical potential is easier in holographic models than baryon chemical potential: large N c baryons much heavier than at finite N c mesons closer to the real-world baryons complicated solitons, mesons elementary fields so far only singular solitons known potentially comparable with the lattice (no sign problem) Bad feature : Artificial, no pure isospin systems exist in nature (weak decays) neutron stars

Chiral Lagrangian At small µ I chiral Lagrangian (with m q = 0 ) to get a feeling what happens L chiral = f 2 4 Tr( D ν UD ν U † ) , π U ∈ U ( N f ) . fπ π a ( x ) T a i U ≡ e T a − − U ( N f ) generators Invariant under separate U → g − 1 L U , U → Ug R The vacuum U = I preserves the vector-like U ( N f ) symmetry, U → g L Ug − 1 g L = g R . R In U = I want to turn on a vector chemical potential µ L = µ R . Other global transformations move us around on the moduli space of vacua, M = U ( N f ) × U ( N f ) U ( N f )

Chiral Lagrangian and µ � = 0 As usual, chemical potentials via D ν U = ∂ ν U − 1 = ∂ ν U − 1 2 δ ν, 0 ( µ L U − Uµ R ) 2 δ ν, 0 ([ µ V , U ] − { µ A , U } ) ( µ L = µ V − µ A , µ R = µ V + µ A ). V χ = f 2 � � π ([ µ V , U ] − { µ A , U } )([ µ V , U † ] + { µ A , U † } ) 4 Tr From V χ minima: ρ A ∼ f 2 (1) µ V = 0 , µ A -any V χ -const. π µ A (2) µ A = 0 , µ V = µ I σ 3 / 2 U max = e iα (cos( β ) I + i sin( β ) σ 3 ) U min = e iα (cos( β ) σ 1 + sin( β ) σ 2 ) and ρ V ∼ f 2 in the U min : π µ I ρ A,I = 0 . � µ 2 A,I < µ 2 U min as in (2) V (3) µ V = µ I σ 3 / 2 , µ A = µ A,I σ 3 / 2 : µ 2 A,I > µ 2 U min opposite V

Vectorial isospin potential Effects of µ V in U = U min ⇔ effects of µ A in U = I vacuum

Aside: non-zero pion mass The chiral Lagrangian gives us the behaviour of Son, Splittorf, Stephanov the pions for small µ I , However, Sakai-Sugimoto has m π = 0 , so we will at small µ I see

Beyond Chiral Langrangian Chiral Langrangian, valid up to the first massive vector meson, µ I ≪ m ρ Other operators are relevant, e.g. Skyrme term 1 �� � 2 � U − 1 ∂ µ U, U − 1 ∂ ν U L Skyrme = 32 e 2 Tr . This leads to a dispersion relation for pions I − k 2 µ 2 − ω 2 + k 2 + µ 2 I = 0 . e 2 f 2 π This suggests massive pions eventually become unstable. But, does not explain what the ρ does. Sakai-Sugimoto has pions and fixed couplings to other mesons. Study π ’s and ρ in this model as function of µ I .

Part III: Part III: Holographic isospin chemical potential Holographic isospin chemical potential

Beyond Chiral Langrangian µ I = 0 Cigar-shaped subspace with D8’s embedded, u = (1 + z 2 ) 1 / 3 No chemical potential no background field, trivial A µ = 0 vacuum. Meson massess from linearised DBI action around trivial vacuum. � A µ ( x µ , z ) = U − 1 ( x ) ∂ µ U ( x ) ψ + ( z ) + B ( n ) µ ( x ) ψ n ( z ) , n ≥ 1 A z = 0 Can go beyond χ -perturbation theory: have χ -Langrangian interacting with infinite tower of massive modes.

Beyond Chiral Langrangian µ I = 0 Effective action we use come from the truncated string effective action � � � u − 1 / 2 γ 1 / 2 Tr( F µν F µν ) + u 5 / 2 γ − 1 / 2 Tr( F µu F µu ) S = ˜ d 4 x d u T + ... where ignored DBI corrections to the YM, ( ( l 2 s F ) n ) and beyond O ( l 3 s ∂F ) For eg., just for pion this gives F zµ = U − 1 ∂ µ U φ (0) ( z ) + B-stuff � � F µν = [ U − 1 ∂ µ U, U − 1 ∂ ν U ] ψ + ( z ) ψ + ( z ) − 1 + B-stuff . which gives chiral Lagrangian plus Skyrme term, � � f 2 � 1 4 ( U − 1 ∂ µ U ) 2 + � � 2 d 4 x Tr π U − 1 ∂ µ U, U − 1 ∂ ν U + “ π ↔ B ′′ S = 32 e 2 1 e 2 ∼ f 2 π ∼ λN c M 2 KK , , . λN c

Sakai-Sugimoto and chiral symmetry In Sakai-Sugimoto, global symmetry is realised as large gauge transformation of bulk field, A µ → gA µ g − 1 + ig∂ µ g − 1 z →−∞ g ( z, x µ ) = g L ∈ SU ( N f ) L , lim z → + ∞ g ( z, x µ ) = g R ∈ SU ( N f ) R . lim

Sakai-Sugimoto and chiral symmetry And changes holonomy � ∞ d z A z ) → g L g − 1 U = P exp ( i R . −∞ changes the pion expectation value, since U = exp ( iπ a ( x ) σ a /f π ) . So if start with trivial vacuum A µ = A z = 0 , the vectorial transformation g L = g R preserves vaccum, does not change U If g L � = g R , does not preseve vacuum i.e. changes holonomy χ -symmetry breaking

Turning on µ I � = 0 For SS model, bulk field A µ ( x, u ) � � � � 1 + O (1 1 + O (1 + ρ ν ( x ) u − 3 / 2 A ν ( x, u ) → B ν ( x ) u ) u ) . here ( R d 4 xB µ J ν ( x ) ) B µ ( x ) ↔ source term for gauge theory current J ν ( x ) ρ ν ( x ) ↔ vev of J µ To add vectorial/axial chemical potential, solve for the even/odd bulk field with b.c. : A µ ( x, z → −∞ ) = µ L δ µ, 0 A µ ( x, z → + ∞ ) = µ R δ µ, 0