This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program

Transcription

1 This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF), under the grants schemes Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes and the program Thales 1

2 Gauge/Gravity Duality 2013 Munich, 29 July 2013 Holography and the Chern-Simons diffusion rate University of Crete APC, Paris 2-

3 Bibliography Based on recent work with: Umut Gursoy, (Utrecht), Ioannis Iatrakis (Crete), Francesco Nitti, (APC), Andy O Bannon (Cambridge) arxiv: [hep-ph] and based also on past work with: Umut Gursoy, (Utrecht), Liuba Mazzanti (Utrecht), Francesco Nitti, (APC) arxiv: [hep-th] arxiv: [hep-th] 3

4 Introduction: Instantons Instantons are important topological semiclassical configurations of SU(N c ) YM theory. They are responsible for the existence of an infinite number of degenerate vacua, and a new coupling constant (the instanton angle θ that breaks the CP symmetry in YM). 4

5 When they can be treated as a dilute instanton gas, their contributions are exponentially small in perturbation theory, e 8π2 Nc λ However, instantons have a size, and large instantons are affected by the IR coupling of the YM Theory that is strong. This is the reason that, although we know how to calculate with individual instantons, the dynamical contributions of instantons to many YM processes are un-calculable. 4-

6 Instantons at large N c Because the instanton factor e 8π2 Nc λ is exponentially suppressed with N c, instanton effects should be exponentially small in the large-n c limit. Veneziano-Witten, solving the η -puzzle, pointed out that this is sometimes false. In QCD, at T=0, the instanton charge is effectively continuous, the instantons cannot be treated like a gas (because large instantons dominate), and instanton effects are NOT exponentially suppressed, but only power suppressed, like other dynamical effects. Witten 79, Veneziano, 79 In particular the mass of the η (the 9-th would-be Goldstone boson of U(1) A in QCD) is M η N f N c Λ QCD 5

7 The U(1) A anomaly The most important role of instantons is to violate the U(1) A charge conservation in QCD J µ 5 = N f i=1 ψ i γ µ γ 5 ψ i µ J µ 5 = N f 16π 2 ϵµνρσ TrF µν F ρσ = N f 8π 2 TrF F. A related number is the Chern-Simons number N CS that characterizes distinct vacua of SU(N c ) YM which cannot be connected with small gauge transformations. It is defined at fixed time, spatial (3d) slices as N CS 1 [ 8π 2 M d3 x ϵ ijk Tr A i j A k 2ig ] 3 A ia j A k where i, j, k = 1, 2, 3. 6

8 The CS diffusion rate The Chern-Simon diffusion rate, Γ CS, is the rate of change of N CS per unit 4-volume and is given by the two-point function of q(x µ ), Γ CS ( N CS) 2 = d 4 x q(x µ )q(0) V t symmetric In equilibrium states with finite temperature, Γ CS is given in terms of G R (ω, k)= Fourier transform of the retarded Green function of q(x µ ) by: Γ CS = lim ω 0 2T ω Im G R (ω, k = 0), Single instanton background contributions to Γ CS are exponentially suppressed. Γ CS can be generated by thermal fluctuations in finite temperature states. Those excite sphaleron configurations which produce non-zero Γ CS upon decay. Since q(x) is a total derivative, Γ CS is identically zero in perturbation theory. A finite value for Γ CS in QCD, signals the creation of net chirality bubbles because of the anomaly of the axial current. These are domains of more left-handed than right-handed quarks or the opposite. 7

9 The chiral magnetic effect The electric current generates a magnetic field, B γze b R (10 19 ) at RHIC (LHC). Or eb 5 15 m 2 π. In neutron stars B Gauss. In magnetars, B Gauss 8

10 A magnetic field, separates spatially the electric charge of left-moving fermions (blue is spin, brown is momentum). Fluctuations of axial charge due to sphalerons, and the strong magnetic field, will generate, charge asymmetry on an event-by-event basis. 8-

11 What is known about Γ CS? Γ CS is a crucial ingredient for the Chiral magnetic effect. The bigger it is, the bigger are the fluctuations of the chiral asymmetry. Γ CS is a non-perturbative, (Minkowskian) transport coefficient. At high enough temperature, using classical field dynamics, hard thermal loop resummation and (and Bödeker s effective theory). It is reliable for (α s 1, 1) 1 log 1 αs Γ CS = 0.21 N ( cg 2 T log m ) D N 2 γ c 1 (N c α s ) 5 T 4 m 2 D = 2N c + N f 6 m 2 D g 2 T 2, γ = N cg 2 T 4π ( N 2 c log m ) D γ , a s = g2 4π Giudice+Shaposhnikov, 93, Moore, 97, 00, Moore+Tassler, 10 N = 4 sym calculation Son+Starinets, 02 Γ CS = λ2 2 8 π 3 T 4 What is Γ CS in YM? 9

12 IHQCD IHQCD is a specially chosen 5d Einstein dilaton model (with two phenomenological parameters in V (λ)) S = M 3 p N 2 c d 5 x g where λ = e ϕ and (M p l) 3 = 45π 2. [ R 4 ( λ) 2 3 λ 2 + V (λ) ] Gursoy+Kiritsis+Nitti, 07 V (λ) = 12 l n=1 V n λ n, λ 0 V (λ) λ 4 3 log λ +, λ The model reproduces correctly the spectra of 0 ++ and 2 ++ glueballs, as well as the finite temperature thermodynamics. It has confinement, a mass gap and asymptotically linear trajectories: m 2 n n. 10

13 The instanton density in IHQCD The instanton density q(x) is dual to an axion field, a(x, r). In N=4 sym this is the usual IIB axion in ten dimensions. The most general action for the axion compatible with the symmetries of the instanton density is S a = M p 3 d 5 x ( ) g Z(λ) ( a) 2 ( a) 4 + O 2 There is no axion potential and therefore the symmetry a a + constant is exact. S a is of order O(Nc 2 ) compared with the IHQCD action. UV λ 0 Z(λ) = Z n=1 N 2 c c n λ n IR λ Z(λ) c 4 λ 4 +, lim n m 2 n (0 + ) m 2 n (0++ ) = 1 Gursoy+Kiritsis+Mazzanti+Nitti, 09 11

14 The θ flow The axion background solution a(r) can be interpreted as a running θ-angle This is in accordance with the absence of UV divergences (all correlators T r[f F ] n are UV finite), and Seiberg-Witten type solutions. The equation of motion is ä + ( 3Ȧ + Ż(λ) Z(λ) ) ȧ = 0, ds 2 = e 2A(r) (dr 2 + dx µ dx µ ) The metric A(r) and λ(r) are taken from the leading order solution. The full solution is a(r) = θ UV + 2πk + C r dre 3A 0 Z(λ), C = q(x) = 1 16π2 T r[f F ] a(r) is a running effective θ-angle. Its running is non-perturbative, a(r) r 4 e b 4 0 λ 12

15 The vacuum energy is r= E(θ UV ) = M p 3 d 5 x g Z(λ) ( a) 2 = M p Ca(r) r=0 Consistency with the θ θ symmetry of YM requires to impose that a( ) = 0. This determines the solution Witten, 79 C = q(x) = θ UV + 2πk 0 E(θ UV ) = E IHQCD + M 3 p 2 Min k The topological susceptibility χ is given by E(θ) = N 2 c E χ θ2 + O ( θ 4 N 2 c ) dr e 3A Z(λ) (θ UV + 2πk) 2 0 dr e 3A Z(λ), χ = 0 Mp 3 dr e 3A Z(λ) The simplest parametrization of Z(λ) consistent with asymptotics is Z(λ) = Z 0 ( 1 + c4 λ 4) Z 0 can be determined from the topological susceptibility (lattice, χ (191MeV) 4 ), and c 4 from the lowest 0 + glueball mass (lattice, m 0 + m 0 ++ = 1.50). The predicted next mass agrees well with lattice (m 0 + m 0 ++ = 2.11). 12-

16 The 2-point function of q(x) We now proceed to calculate the 2-point function q(x)q(0) at finite temperature, T T c. The linear fluctuation equation of the axion ((in Fourier space) is considered in the black hole background, ds 2 = e 2A dr 2 f fdt2 + dx i dx i ), [ 1 Z(λ) g r ( Z(λ) g g rr r ) g µν k µ k ν ] with in-going wave boundary conditions at the horizon. The on shell action of the fluctuation is S on-shell α = d 4 k (2π) 4 a( kµ ) F(r, k µ ) a(k µ ) δα(r, k µ ) = 0, F(r, k µ ) M 3 p δα(r, kµ ) Z(λ) g g rr r δα(r, k µ )/2 The AdS/CFT dictionary yields that the retarded Green s function is Ĝ R (ω, k) = 2 lim r 0 F(r, k µ ). 0 r h, 13

17 The calculation of the CS diffusion rate We can calculate directly the IR limit of G R and therefore Γ CS to be Γ CS st/nc 2 = Z(λ h) 2π, which has implicit dependence on T through Z(λ h ). λ h is the value of the dilaton at the black hole horizon. On the large black hole branch, Γ CS /(st/n 2 c ) is bounded from below by its value in the T limit,. CS Κ 2 Z 0 2Π s T N c 2 lim T Γ CS st/n 2 c = Z 0 2π T T c 14

18 The previous calculation of Γ CS is not reliable because: (a) The parametrization was crude (b) λ h < 1 The largest polynomial correction to Z(λ) for small values of λ will come from linear terms so we reparametrize Z(λ) as Z(λ) = Z 0 ( 1 + c1 λ + c 4 λ 4) To constrain c 1 we will demand that our holographic results for the axial glueball masses fall within one σ of the lattice values for the two first 0 + glueball masses 0 < c 1 < 5, 0.06 < c 4 < 50 Simple non-monotonic Z(λ)s with the desired asymptotics, produce glueball masses that deviate from lattice results more than 10%. This implies that the bound Γ CS st/n 2 c Z 0 2π is always valid. 14-

19 IHQCD masses of the 0 + glueball states with excitation number n, normalized to the 0 ++ mass for various (c 1, c 4 ). From the top (red) dots to the bottom (blue) dots: (c 1, c 4 ) = (0, 0.26), (0.5, 0.87), (1, 2.2), (5, 24), (10, 75 The two horizontal blue lines with surrounding blue bands indicate the results and errors, respectively, of the large-n YM lattice masses of the lowest and first excited states, n = 1 and n = 2, (Morningstar-Peardon). 14-

20 CS Κ 2 Z 0 2Π s T N c 2 8 CS Κ 2 Z 0 2Π s T N c T T c 0.80, T T c (a) The numerical result for Γ CS /(st/n 2 c ), normalized to the T value κ 2 Z 0 /2π, as functions of T/T c, for different values of the parameters (c 1, c 4 ). From the bottom (red) curve to the top (blue) curve, (c 1, c 4 ) = (0, 0.26), (0.5, 0.87), (1, 2.2), (5, 24), (10, 75), (20, 230), (40, 600). (b) Close-up of the curves for (from bottom to top) (c 1, c 4 ) = (0, 0.26), (0.5, 0.87), (1, 2.2). 14-

21 A numerical estimate of Γ CS near the confinement-deconfinement phase transition is Γ CS (T c ) T 4 c = 0.31 Z(λ c) 2π > 0.31 Z 0 2π = 1.64, where we have used the lattice result for the entropy density, s(t c ) = 0.31N 2 c T 3 c, (Lucini-Teper-Wenger). At 1 σ an upper bound can also be set to Γ CS (T c ). In total 1.64 Γ CS(T c ) Tc We may compare this range of values with the N=4 result taking the standard values, λ 6π to get a result that is at least 40 times smaller. Γ N = 4 sym : CS (T ) T A naive extrapolation of the Moore-Tassler high temperature formula to α s = 0.5, gives the right order of magnitude, Γ MT CS (T c) T 4 c = 30α 5 s 1 (1) 14-

22 The CS correlator k 10 T c Ω T c 20 Im G R T c M p Our numerical results for Im G R (ω, k = 0; T c, 2T c )/(T c Mp 3 ) as a function of ω/t c and k /T c. As k increases up to k /T c 10, the largest peak shifts from ω/t c 22 up to ω/t c 30. The width of the peak changes very little. 15

23 Outlook The Chern-Simons diffusion rate seems to be non-negligible in QCD and is bounded below. CP-odd phenomena above T c seem to be controlled by a long lived excitation that has the same order of magnitude mass as the 0 + glueball. Fixing better Z(λ) by the fitting to a lattice calculation of the Euclidean 2-point function of q(x µ ) is an important future plan. Calculate Γ CS in a model which incorporates the flavor degrees of freedom as well (V-QCD). The currents which appear in chiral magnetic effect can be calculated holographically upon addition of the flavor in the pure glue background. In this case, a vacuum state with a magnetic field and net chirality should be constructed. 16

24 . Thank you 17

25 This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF), under the grants schemes Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes and the program Thales 18

26 Changes of N CS The change of N CS during a dynamical process is given by N CS = d 4 x q(x µ ) q(x µ ) 1 16π 2 M Tr [F F ] = 1 32π 2 ϵµνρσ TrF µν F ρσ. We conclude that instantons mediate the changes of axial U(1) charge. N 5 = (N L N R ) t= (N L N R ) t= = 2N f N CS. 19

27 In a thermal plasma, such fluctuations of axial U(1) number are controlled by a Langevin-like process d( N 5 ) = ξ CS N 5 + η(t), η(t)η(t ) Γ CS δ(t t ) dt where Γ CS is the Chern-Simons diffusion rate. As usual, the white-noise correlation is controlled by the symmetric correlator while the friction coefficient ξ CS is controlled by the retarded correlator. Consider now the transition amplitude, A, from a state ψ 1 (t 1 ) to ψ 2 (t 2 ) times the change in axial number A N 5 = 2 d 4 x ψ 2 q(x) ψ 1 Taking the square of the above equation, summing over all possible ψ 2 (t 2 ) and using the completeness relation I = ψ 2 ψ 2 we obtain ( N 5 ) 2 = 4 d 4 x d 4 y ψ 1 q(x) q(y) ψ 1 RETURN 19-

28 The Chern-Simon diffusion rate, Γ CS, is the rate of change of N CS per unit 4-volume and is given by the two-point function of q(x µ ), Γ CS ( N CS) 2 = d 4 x q(x µ )q(0) V t symmetric In equilibrium states with finite temperature, Γ CS is given by Γ CS = 2T ξ CS = lim ω 0 2T ω Im G R (ω, k = 0), where G R (ω, k)= Fourier transform of the retarded Green function of q(x µ ). Single instanton background contributions to Γ CS are exponentially suppressed. Γ CS can be generated by thermal fluctuations in finite temperature states. Those excite sphaleron configurations which produce non-zero Γ CS upon decay. Since q(x) is a total derivative, Γ CS is identically zero in perturbation theory. Finite Γ CS in QCD, signals the creation of net chirality bubbles (domains of more lefthanded than right-handed quarks or the opposite) because of the anomaly of the axial current. 19-

29 Θ The θ RG flow 1.0 Θ UV E MeV The effective θ vanishes in the IR. q(r) is a marginally-irrelevant operator. We have taken: Z(λ) = Z 0 (1 + c 4 λ 4 ) ( λ4 ) The effective θ-angle runs also in the D4 model for QCD, and also vanishes in the IR θ(u) = θ(1 U 3 0 /U 3 ) 20

30 The CS spectral function We now calculate the axion spectral function ImG R (ω, k) for non-zero ω and k. We first set k = 0 For small ω, the axion fluctuation equation yields Im G R (ω, k = 0) ω. For large ω, we expect Im G R (ω, k = 0) ω 4 because the theory is conformally invariant in the UV and q(x µ ) has dimension four. 21

31 Im G R T c M p Ω T c Im G R (ω, k = 0)/(T c M 3 p ) as a function of ω/t c, at T c. The red dots are our numerical results that match the solid blue curve, ( ) (ω/t c ) 4.051, which is the expected large ω behavior of the spectral function. 21-

32 The ω 4 scaling of Im G R (ω, k = 0) at large ω comes from the UV (free) part of the two-point function. It overwhelms peaks in Im G R (ω, k), rendering them practically invisible. To eliminate the large-ω divergence we compute G R (ω, k) at two different temperatures, T 1 and T 2, and then take the difference, G R (ω, k; T 1, T 2 ) G R (ω, k) G T2 R (ω, k). T1 21-

33 Im G R T c M p Ω T c The difference Im G R (ω, k = 0; T c, 2T c )/(T c M 3 p ) as a function of ω/t c. The difference goes to zero as ω/t c. The prominent minimum at ω/t c 10 and maximum at ω/t c 22 indicate a shift in spectral weight with increasing T, possibly from the motion of a peak in the spectral function. There is a peak at ω 20T c 1600MeV and width 10T c 1300MeV. 21-

34 Comparison with lattice data (Meyer) M M n 3000 n (a) (b) Comparison of glueball spectra from our model with b 0 = 4.2, l 0 = 0.05 (boxes), with the lattice QCD data from Ref. I (crosses) and the AdS/QCD computation (diamonds), for (a) 0 ++ glueballs; (b) 2 ++ glueballs. The masses are in MeV, and the scale is normalized to match the lowest 0 ++ state from Ref. I. 22

35 The fit to glueball lattice data J P C Ref I (MeV) Our model (MeV) Mismatch N c Mismat (4%) (5%) % 2153 (10%) 5% (4%) (4%) (12%) 2% (5%) % (4%) % (4%) % (5%) % Comparison between the glueball spectra in Ref. I and in our model. The states we use as input in our fit are marked in red. The parenthesis in the lattice data indicate the percent accuracy. 23

36 Fit and comparison HQCD lattice N c = 3 lattice N c Parameter [p/(n 2 c T 4 )] T =2Tc V 1 = 14 L h /(N 2 c T 4 c ) (Karsch) 0.31 (Teper+Lucini) V 3 = 170 [p/(n 2 c T 4 )] T + π 2 /45 π 2 /45 π 2 /45 M p l = [45π 2 ] 1/3 m 0 ++/ σ (Chen ) 3.37 (Teper+Lucini) l s /l = 0.15 m 0 +/m (Chen ) - c a = 0.26 χ (191MeV ) 4 (191MeV ) 4 (DelDebbio) - Z 0 = 133 T c /m (7) m 0 ++/m (11) 1.90(17) m 2 ++/m (4) 1.46(11) m 0 +/m (10) - 24

37 G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier and B. Petersson, Thermodynamics of SU(3) Lattice Gauge Theory, Nucl. Phys. B 469, 419 (1996) [arxiv:hep-lat/ ]. B. Lucini, M. Teper and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP 0502, 033 (2005) [arxiv:hep-lat/ ]; SU(N) gauge theories in four dimensions: Exploring the approach to N =, JHEP 0106, 050 (2001) [arxiv:hep-lat/ ]. Y. Chen et al., Glueball spectrum and matrix elements on anisotropic lattices, Phys. Rev. D 73 (2006) [arxiv:hep-lat/ ]. L. Del Debbio, L. Giusti and C. Pica, Topological susceptibility in the SU(3) gauge theory, Phys. Rev. Lett. 94, (2005) [arxiv:hepth/ ]. 24-

38 The pressure from the lattice at different N Marco Panero arxiv:

39 The entropy from the lattice at different N Marco Panero arxiv:

40 The trace from the lattice at different N Marco Panero arxiv:

41 The specific heat C v T 3 2 N c T T c 28

42 The speed of sound c s T T c 29

43 Comparing to Gubser+Nelore s formula Gubser+Nelore proposed the following approximate formula for the speed of sound c 2 s V 2 V 2 ϕ=ϕh Gursoy (unpublished) 2009 Red curve=numerical calculation, Blue curve=gubser s adiabatic/approxima formula. 30

44 Detailed plan of the presentation Title page 0 minutes Bibliography 1 minutes Introduction: Instantons 3 minutes Instantons at large N c 4 minutes The U(1) A anomaly 5 minutes The CS diffusion rate 6 minutes The chiral magnetic effect 9 minutes What is known about Γ CS? 11 minutes IHQCD 12 minutes The instanton density in IHQCD 14 minutes The θ-flow 18 minutes The two-point function of q(x) 19 minutes Calculation of the CS diffusion rate 26 minutes The CS correlator 27 minutes 31

45 Outlook 30 minutes The changes of N CS 35 minutes The θ RG flow 36 minutes The CS spectral function 38 minutes Comparison with lattice data (Meyer) 40 minutes The fit to glueball lattice data 42 minutes Fit and comparison 44 minutes The pressure 45 minutes The entropy 46 minutes The trace 49 minutes The free energy 51 minutes The specific heat 56 minutes The speed of sound 59 minutes The Gubser-Nelore formula for c s 65 minutes 31-