Dynamics of heavy quarks in charged N = 4 SYM plasma Aleksi Vuorinen University of Washington, Seattle & Technical University of Vienna C. Herzog and AV, arxiv:0708:0609 [hep-th]
Outline N = 4 SYM and Finite temperatures Motivation AdS/CFT preliminaries N = 4 SYM and QCD Setup Gravity calculation Puzzle with R charges The metric Single charge solutions Results
Motivation AdS/CFT preliminaries N = 4 SYM and QCD Outline N = 4 SYM and Finite temperatures Motivation AdS/CFT preliminaries N = 4 SYM and QCD Setup Gravity calculation Puzzle with R charges The metric Single charge solutions Results
Motivation AdS/CFT preliminaries N = 4 SYM and QCD Why bother? Heavy ion experiments Need quantitative understanding of non-abelian plasmas at High T and small/moderate µ Strong coupling In and (especially) out-of equilibrium Combination notoriously problematic Perturbative methods of limited use Dynamical quantities hard for lattice QCD Any new insights into dynamics of strongly coupled non-abelian plasmas welcome!
Motivation AdS/CFT preliminaries N = 4 SYM and QCD N = 4 SYM and gauge/gravity duality Unique feature of N = 4 SYM: It has known string dual N = 4 SYM IIB string theory on AdS5 S 5 ds 2 = R 2 { dr 2 r 2 + r 2( dt 2 + dx 2) + dω 2 5 } R = (curvature) radius of S5 and AdS 5 Parameters string coupling gs and length scale l s = α AdS/CFT dictionary: Radial coordinate in AdS energy scale in CFT (R/ls ) 4 = λ, g s = λ/(4πn c ) Beautiful aspect: Classical sugra limit: string coupling gs 1 and R/l s 1 Gauge theory at λ 1, Nc 1 Classical GR!
Motivation AdS/CFT preliminaries N = 4 SYM and QCD N = 4 SYM and gauge/gravity duality Unique feature of N = 4 SYM: It has known string dual N = 4 SYM IIB string theory on AdS5 S 5 ds 2 = R 2 { dr 2 r 2 + r 2( dt 2 + dx 2) + dω 2 5 } R = (curvature) radius of S5 and AdS 5 Parameters string coupling gs and length scale l s = α AdS/CFT dictionary: Radial coordinate in AdS energy scale in CFT (R/ls ) 4 = λ, g s = λ/(4πn c ) Beautiful aspect: Classical sugra limit: string coupling gs 1 and R/l s 1 Gauge theory at λ 1, Nc 1 Classical GR!
Motivation AdS/CFT preliminaries N = 4 SYM and QCD N = 4 SYM and gauge/gravity duality Unique feature of N = 4 SYM: It has known string dual N = 4 SYM IIB string theory on AdS5 S 5 ds 2 = R 2 { dr 2 r 2 + r 2( dt 2 + dx 2) + dω 2 5 } R = (curvature) radius of S5 and AdS 5 Parameters string coupling gs and length scale l s = α AdS/CFT dictionary: Radial coordinate in AdS energy scale in CFT (R/ls ) 4 = λ, g s = λ/(4πn c ) Beautiful aspect: Classical sugra limit: string coupling gs 1 and R/l s 1 Gauge theory at λ 1, Nc 1 Classical GR!
Motivation AdS/CFT preliminaries N = 4 SYM and QCD AdS/CFT dictionary at finite temperature Turning on finite temperature on gauge theory side Adding a black hole in AdS 5, with horizon at radial coordinate value r h = πt Chemical potentials for SO(6) R-symmetry Black hole spinning in S 5 directions
Motivation AdS/CFT preliminaries N = 4 SYM and QCD AdS/CFT dictionary at finite temperature Turning on finite temperature on gauge theory side Adding a black hole in AdS 5, with horizon at radial coordinate value r h = πt Chemical potentials for SO(6) R-symmetry Black hole spinning in S 5 directions Adding massive N = 2 hypermultiplet Adding a D7-brane wrapping AdS 5 down to some r 0 Quarks and mesons Classical strings hanging from D7 brane
Motivation AdS/CFT preliminaries N = 4 SYM and QCD What is N = 4 SYM? Maximally supersymmetric Yang-Mills theory with 4 SUSY generators Field content: 1 gauge field, 3 complex scalars, 4 Majorana fermions all massless, as theory conformal Relevant symmetries: local SU(N c ) gauge symmetry (all fields in adj. repr.) and global SO(6) R symmetry
Motivation AdS/CFT preliminaries N = 4 SYM and QCD What is N = 4 SYM? Maximally supersymmetric Yang-Mills theory with 4 SUSY generators Field content: 1 gauge field, 3 complex scalars, 4 Majorana fermions all massless, as theory conformal Relevant symmetries: local SU(Nc ) gauge symmetry (all fields in adj. repr.) and global SO(6) R symmetry N = 4 SYM vs. QCD: T = 0 Conformally invariant: No Λ QCD, fixed coupling No S-matrix, no particles Conclusion: No real similarity
Motivation AdS/CFT preliminaries N = 4 SYM and QCD What is N = 4 SYM? Maximally supersymmetric Yang-Mills theory with 4 SUSY generators Field content: 1 gauge field, 3 complex scalars, 4 Majorana fermions all massless, as theory conformal Relevant symmetries: local SU(N c ) gauge symmetry (all fields in adj. repr.) and global SO(6) R symmetry N = 4 SYM vs. QCD: T 0 Deconfined non-abelian plasma Similar IR physics: Debye screening, finite magnetic mass Similar hydro properties Conclusion: Near perfect qualitative match!
Motivation AdS/CFT preliminaries N = 4 SYM and QCD Plan of attack Challenge 1: Obtain as much insight into strongly coupled N = 4 SYM as possible Especially dynamical quantities inaccessible in strongly coupled QCD interesting An enormous activity during past few years Challenge 2: Interpolate to moderate couplings Compute NLO 1/λ corrections at strong coupling Investigate weak coupling limit of N = 4 SYM Challenge 3: Attempt to infer information on QCD Couplings relevant for heavy ion physics in no man s land No clear prescription, but a lot of room to play around... Holy Grail: Find string/gravity dual of QCD
Motivation AdS/CFT preliminaries N = 4 SYM and QCD Plan of attack Challenge 1: Obtain as much insight into strongly coupled N = 4 SYM as possible Especially dynamical quantities inaccessible in strongly coupled QCD interesting An enormous activity during past few years Challenge 2: Interpolate to moderate couplings Compute NLO 1/λ corrections at strong coupling Investigate weak coupling limit of N = 4 SYM Challenge 3: Attempt to infer information on QCD Couplings relevant for heavy ion physics in no man s land No clear prescription, but a lot of room to play around... Holy Grail: Find string/gravity dual of QCD
Motivation AdS/CFT preliminaries N = 4 SYM and QCD Plan of attack Challenge 1: Obtain as much insight into strongly coupled N = 4 SYM as possible Especially dynamical quantities inaccessible in strongly coupled QCD interesting An enormous activity during past few years Challenge 2: Interpolate to moderate couplings Compute NLO 1/λ corrections at strong coupling Investigate weak coupling limit of N = 4 SYM Challenge 3: Attempt to infer information on QCD Couplings relevant for heavy ion physics in no man s land No clear prescription, but a lot of room to play around... Holy Grail: Find string/gravity dual of QCD
Motivation AdS/CFT preliminaries N = 4 SYM and QCD Plan of attack Challenge 1: Obtain as much insight into strongly coupled N = 4 SYM as possible Especially dynamical quantities inaccessible in strongly coupled QCD interesting An enormous activity during past few years Challenge 2: Interpolate to moderate couplings Compute NLO 1/λ corrections at strong coupling Investigate weak coupling limit of N = 4 SYM Challenge 3: Attempt to infer information on QCD Couplings relevant for heavy ion physics in no man s land No clear prescription, but a lot of room to play around... Holy Grail: Find string/gravity dual of QCD
Setup Gravity calculation Puzzle with R charges Outline N = 4 SYM and Finite temperatures Motivation AdS/CFT preliminaries N = 4 SYM and QCD Setup Gravity calculation Puzzle with R charges The metric Single charge solutions Results
Setup Gravity calculation Puzzle with R charges Heavy quarks in strongly coupled N = 4 SYM Herzog et al, hep-th/0605158;... Consider dragging a heavy quark of a fundamental N = 2 hypermultiplet through strongly interacting N = 4 SYM plasma Want to compute friction coefficient µ in eom dp dt = µp + f Dragging the quark at constant velocity, f = Mvµ Letting the quark slow down after initial kick, p(t) = p(0)e µt
Setup Gravity calculation Puzzle with R charges Heavy quarks in strongly coupled N = 4 SYM Herzog et al, hep-th/0605158;... Consider dragging a heavy quark of a fundamental N = 2 hypermultiplet through strongly interacting N = 4 SYM plasma Want to compute friction coefficient µ in eom dp dt = µp + f Dragging the quark at constant velocity, f = Mvµ Letting the quark slow down after initial kick, p(t) = p(0)e µt
Setup Gravity calculation Puzzle with R charges Gravity computation a la Herzog et al. Start from classical Nambu-Goto action S = 1 2πα dσ dτ det G And solve for classical string profile with correct bc s x(r, t) = x 0 + vt + v { π 2r h 2 arctan r arccoth r } r h r h
Setup Gravity calculation Puzzle with R charges Gravity computation a la Herzog et al. Start from classical Nambu-Goto action S = 1 2πα dσ dτ det G with G induced world-sheet (τ = t, σ = r) metric obtained from black hole line element { dr ds 2 = L 2 2 h h dt2 + r 2 dx }, ( 2 h = r 2 1 (r h /r) 4) And solve for classical string profile with correct bc s x(r, t) = x 0 + vt + v { π 2r h 2 arctan r arccoth r } r h r h
Setup Gravity calculation Puzzle with R charges Gravity computation a la Herzog et al. Start from classical Nambu-Goto action S = 1 2πα dσ dτ det G with G induced world-sheet (τ = t, σ = r) metric obtained from black hole line element { dr ds 2 = L 2 2 h h dt2 + r 2 dx }, ( 2 h = r 2 1 (r h /r) 4) And solve for classical string profile with correct bc s x(r, t) = x 0 + vt + v { π 2r h 2 arctan r arccoth r } r h r h
Setup Gravity calculation Puzzle with R charges Result for friction and diffusion coefficients dp = π λt 2 v dt 2, 1 v 2 µ = π λt 2, 2 M kin T 2 D = µm kin π λt This may be compared to weak coupling expansion { 12π D = d A g 4 ln 2T + 13 T m D 12 γ E + 1 } 3 ln 2 + ζ (2) ζ(2) P. Chesler and AV, hep-ph/0607148
Setup Gravity calculation Puzzle with R charges Result for friction and diffusion coefficients dp = π λt 2 v dt 2, 1 v 2 µ = π λt 2, 2 M kin T 2 D = µm kin π λt This may be compared to weak coupling expansion { 12π D = d A g 4 ln 2T + 13 T m D 12 γ E + 1 } 3 ln 2 + ζ (2) ζ(2) P. Chesler and AV, hep-ph/0607148
Setup Gravity calculation Puzzle with R charges
Setup Gravity calculation Puzzle with R charges Can also compute energy perturbation in the plasma created by the heavy quark; P. Chesler and L. Yaffe, arxiv:0706.0368 [hep-th]
Setup Gravity calculation Puzzle with R charges Puzzle with R charges To model finite u and d quark chemical potentials in QCD, want to investigate finite R charge µ in N = 4 SYM In the literature, there exist two mutually incompatible solutions Herzog s (H; hep-th/0605191) 5d solution with velocity dependent µ Caceres and Guijosa s (CG; hep-th/0605235) simplified 10d solution with velocity independent µ Obvious questions: Does H uplift to 10d? If yes, what is its relation to CG on the SYM side? Would be interesting to find all relevant 10d solutions...
Setup Gravity calculation Puzzle with R charges Puzzle with R charges To model finite u and d quark chemical potentials in QCD, want to investigate finite R charge µ in N = 4 SYM In the literature, there exist two mutually incompatible solutions Herzog s (H; hep-th/0605191) 5d solution with velocity dependent µ Caceres and Guijosa s (CG; hep-th/0605235) simplified 10d solution with velocity independent µ Obvious questions: Does H uplift to 10d? If yes, what is its relation to CG on the SYM side? Would be interesting to find all relevant 10d solutions...
Setup Gravity calculation Puzzle with R charges Puzzle with R charges To model finite u and d quark chemical potentials in QCD, want to investigate finite R charge µ in N = 4 SYM In the literature, there exist two mutually incompatible solutions Herzog s (H; hep-th/0605191) 5d solution with velocity dependent µ Caceres and Guijosa s (CG; hep-th/0605235) simplified 10d solution with velocity independent µ Obvious questions: Does H uplift to 10d? If yes, what is its relation to CG on the SYM side? Would be interesting to find all relevant 10d solutions...
The metric Single charge solutions Results Outline N = 4 SYM and Finite temperatures Motivation AdS/CFT preliminaries N = 4 SYM and QCD Setup Gravity calculation Puzzle with R charges The metric Single charge solutions Results
The metric Single charge solutions Results The spinning black D3-brane background ds 2 10 = ds 2 5 + R2 3 i=1 ( X 1 i dµ 2 i + µ 2 i (dψ i + A i /R) 2), ds 2 5 = (H 1 H 2 H 3 ) 2/3 h dt 2 + (H 1 H 2 H 3 ) 1/3 (h 1 dr 2 + r 2 X i = H 1 i (H 1 H 2 H 3 ) 1/3 ; A i = h = m r 2 + r 2 R 2 H 1H 2 H 3 ; = (1 H 1 i ) m 3 i=1 l i dt ; R 2 dx2 ) ; X i µ 2 i ; H i = 1 + l2 i r 2 ; µ 1 = sin θ 1 ; µ 2 = cos θ 1 sin θ 2 ; µ 3 = cos θ 1 cos θ 2.
The metric Single charge solutions Results Field theory temperature and chemical potentials: 1 2rh 6 T = + (l2 1 + l2 2 + l2 3 )r h 4 l2 1 l2 2 l2 3 2πrh 2R2, 3 i=1 rh 2 + l2 i Φ i = Ai R Ai r R = l 3 i j=1 rh 2 + l2 i r=rh R 2 r h (rh 2 + l2 i ) Challenge again: Solve classical equations of motion for strings stretching from D7 brane to horizon Boundary conditions at D7 brane and dependence on angular coordinates important for field theory interpretation From now on, for simplicity consider turning on only one chemical potential
The metric Single charge solutions Results Field theory temperature and chemical potentials: 1 2rh 6 T = + (l2 1 + l2 2 + l2 3 )r h 4 l2 1 l2 2 l2 3 2πrh 2R2, 3 i=1 rh 2 + l2 i Φ i = Ai R Ai r R = l 3 i j=1 rh 2 + l2 i r=rh R 2 r h (rh 2 + l2 i ) Challenge again: Solve classical equations of motion for strings stretching from D7 brane to horizon Boundary conditions at D7 brane and dependence on angular coordinates important for field theory interpretation From now on, for simplicity consider turning on only one chemical potential
The metric Single charge solutions Results Single charge solutions Introduction of chemical potentials breaks SO(6) R symmetry into SO(4). Turning on µ for Two elements of Cartan sub-algebra of SO(6) in unbroken SO(4) D7 brane and string neutral One element in broken part D7 brane and string charged First case corresponds to polar, second to equatorial strings
The metric Single charge solutions Results Single charge solutions Introduction of chemical potentials breaks SO(6) R symmetry into SO(4). Turning on µ for Two elements of Cartan sub-algebra of SO(6) in unbroken SO(4) D7 brane and string neutral One element in broken part D7 brane and string charged First case corresponds to polar, second to equatorial strings
The metric Single charge solutions Results Various possibilities for the D7 brane the string is attached to Non-rotating brane not extending in ψ directions ψ = 0 Non-rotating brane extending in ψ directions No torque condition at brane, Neumann bc s Rotating brane
The metric Single charge solutions Results Various possibilities for the D7 brane the string is attached to Non-rotating brane not extending in ψ directions ψ = 0 Non-rotating brane extending in ψ directions No torque condition at brane, Neumann bc s Rotating brane Simplest case with known stable D7 brane configuration; CG solution On field theory side, seems to correspond to non-zero R charge with zero chiral condensate Possible problem: For charged string, R charge density diverges
The metric Single charge solutions Results Various possibilities for the D7 brane the string is attached to Non-rotating brane not extending in ψ directions ψ = 0 Non-rotating brane extending in ψ directions No torque condition at brane, Neumann bc s Rotating brane Intuitive gravitational picture; uplift of H solution On the field theory side, seems to correspond to non-zero chiral condensate Problem: At T = 0, brane configuration unstable Too cool to be physical...
The metric Single charge solutions Results
The metric Single charge solutions Results Various possibilities for the D7 brane the string is attached to Non-rotating brane not extending in ψ directions ψ = 0 Non-rotating brane extending in ψ directions No torque condition at brane, Neumann bc s Rotating brane No known examples of suitable classical brane solutions Field theory interpretation includes spatial gradient in R charge chemical potential
The metric Single charge solutions Results Results I General result for string profile in terms of forces in x and ψ directions, P 2πα πx 1 and L 2πα πψ 1 (x ) 2 = βɛ ( ±Lvγ(φ + ω) + P( α + v 2 γ) ) 2 αγ γɛ(l(φ + ω) ± Pv) 2 L 2 f 2 αγ P 2, αɛ αγɛi 1 (ψ ) 2 = βγ ( ( ±L ɛ(φ + ω) 2 f 2 α ) + Pvɛ(φ + ω) ) 2 αɛ γɛ(l(φ + ω) ± Pv) 2 L 2 f 2 αγ P 2 αɛ αγɛi 1 CG solution with ω ψ = 0, H with L = 0 α, β, etc. complicated functions of r, rh, l, R P from requiring x, ψ be well-defined from r h to r 0 Both numer. and denomin. vanish at some r = r 1 (r h, l, R)
The metric Single charge solutions Results Results I General result for string profile in terms of forces in x and ψ directions, P 2πα πx 1 and L 2πα πψ 1 (x ) 2 = βɛ ( ±Lvγ(φ + ω) + P( α + v 2 γ) ) 2 αγ γɛ(l(φ + ω) ± Pv) 2 L 2 f 2 αγ P 2, αɛ αγɛi 1 (ψ ) 2 = βγ ( ( ±L ɛ(φ + ω) 2 f 2 α ) + Pvɛ(φ + ω) ) 2 αɛ γɛ(l(φ + ω) ± Pv) 2 L 2 f 2 αγ P 2 αɛ αγɛi 1 CG solution with ω ψ = 0, H with L = 0 α, β, etc. complicated functions of r, rh, l, R P from requiring x, ψ be well-defined from r h to r 0 Both numer. and denomin. vanish at some r = r 1 (r h, l, R)
The metric Single charge solutions Results Results II Final result: Drag force πx 1 = 1 2πα g xx (r 1 )v All solutions investigated (CG, H) appear stable against small perturbations Full analysis awaits Polar CG solutions leave door open for super-radiant instabilities Main lesson: Drag force depends strongly on transverse (S 5 ) geometry Bad news for hopes to obtain information relevant for QCD Is R charge chemical potential at all anaus to (light) quark µ in QCD?
The metric Single charge solutions Results Results II Final result: Drag force πx 1 = 1 2πα g xx (r 1 )v All solutions investigated (CG, H) appear stable against small perturbations Full analysis awaits Polar CG solutions leave door open for super-radiant instabilities Main lesson: Drag force depends strongly on transverse (S 5 ) geometry Bad news for hopes to obtain information relevant for QCD Is R charge chemical potential at all anaus to (light) quark µ in QCD?
The metric Single charge solutions Results Results II Final result: Drag force πx 1 = 1 2πα g xx (r 1 )v All solutions investigated (CG, H) appear stable against small perturbations Full analysis awaits Polar CG solutions leave door open for super-radiant instabilities Main lesson: Drag force depends strongly on transverse (S 5 ) geometry Bad news for hopes to obtain information relevant for QCD Is R charge chemical potential at all anaus to (light) quark µ in QCD?
Outline N = 4 SYM and Finite temperatures Motivation AdS/CFT preliminaries N = 4 SYM and QCD Setup Gravity calculation Puzzle with R charges The metric Single charge solutions Results
Summary and conclusions AdS/CFT duality unique and powerful tool for studying strongly coupled non-abelian plasmas Dynamical properties of N = 4 SYM theory available at large N c, λ limit Recently much progress in gravity duals of other related theories Hope to eventually learn lessons about QGP physics At finite T, SYM and QCD plasmas qualitatively similar Quantitative predictions presently questionable; weak coupling calculations important step in this direction Discrepancy in drag force of heavy quarks in charged N = 4 plasma solved CG solution seems stable and physically simplest Strong dependence of results on transverse geometry