The Phases of QCD in the T, N f Plane

Transcription

1 Extreme QCD 2008 Raleigh, North Carolina State University, July The Phases of QCD in the T, N f Plane Maria Paola Lombardo Istituto Nazionale di Fisica Nucleare Laboratori Nazionali di Frascati Roma, Italia In collaboration with Albert Deuzmann and Elisabetta Pallante [hep-lat] Talks by E. Pallante and A. Deuzemann at Lattice08 - Beyond QCD Session. MpL - QCD in the T,Nf plane 1

2 PROPERTIES: Chiral Symmetry Screening, confinement Asymptotic freedom Scale Invariance MpL - QCD in the T,Nf plane 2

3 OUTLINE: The Banks-Zacs scenario The conformal window Early Lattice Studies Introducing temperature Results Discussions MpL - QCD in the T,Nf plane 3

4 HISTORY : THE BANKS ZACS scenario The Lagrangian of an SU(N) gauge theory is: L = ψ(i + g(µ) A a T a )ψ F a µν F aµν The renormalization group (RG) equation for the running coupling is: µ µ α(µ) = β(α) b α2 (µ) c α 3 (µ) d α 4 (µ)..., where α(µ) = g 2 (µ)/4π. With the N f fermions in the fundamental representation c = 1 24π 2 ( b = 1 6π (11N 2N f) 34N 2 10NN f 3 N 2 1 N N f ). MpL - QCD in the T,Nf plane 4

5 The theory is asymptotically free if b > 0 : N f < 11 2 N At two loops, the theory has an infrared stable, non-trivial fixed point if b > 0 and c < 0. 34N 3 13N < N f < 11 2 N N f < N f < N f In this case the fixed point is at α = b c. The coefficients b and c are scheme-independent : if a zero, α, of the β function exists at two loops, it exists to any order in perturbation theory. MpL - QCD in the T,Nf plane 5

6 Nf N** f S N* f A A 2 g(0) PHASES OF QCD -- BANKS ZACS MpL - QCD in the T,Nf plane 6

7 THE DISCOVERY OF THE CONFORMAL WINDOW OF QCD Appelquist et al CHIRAL SYMMETRY BREAKING NEEDS LARGE GAUGE COUPLING: Chiral symmetry breaking is possible when the gauge coupling exceeds a critical value π α c 3 C 2 (R) = 2π N 3 (N 2 1), where C 2 (R) is the quadratic Casimir of the the representation R. Thus we would expect that when N f is decreased below the value N c f at which α = α c, the theory undergoes a transition to a phase where chiral symmetry is spontaneously broken. N c f = N ( 100N 2 ) 66 25N MpL - QCD in the T,Nf plane 7

8 N f N ** f S N cr f S A N * f A 2 g (0) cr PHASES OF QCD - THE CONFORMAL WINDOW g (0) MpL - QCD in the T,Nf plane 8

9 Nf Nf N** f S ** Nf S cr Nf S A N* f A * Nf A A 2 2 PHASES OF QCD -- BANKS ZACS g(0) g(0) cr PHASES OF QCD - THE CONFORMAL WINDOW g(0) MpL - QCD in the T,Nf plane 9

10 Phases of QCD in the N f, N c plane D. Dietrich, F. Sannino 2006; D. Dietrich xqcd07 MpL - QCD in the T,Nf plane 10

11 C-2+>) C&7.+)1. += +' 9>B/#(/H/#''': P P +2= N= P 9&:+' ++;9= :7;9= : =+ ( +.;0' >/2= O= P # = P. /++ ;9= :7;9= : +. /..7+ 9&:+++ From R. Brower MpL - QCD in the T,Nf plane 11

12 ONE MORE DIMENSION : TEMPERATURE Qualitative graph of the phase diagram in the T, N f plane. MpL - QCD in the T,Nf plane 12

13 QUANTITATIVE ESTIMATES OF THE CHIRAL BOUNDARY FROM RG direction 3-direction α s (k,t) 1 T cr [Mev] k [GeV] N f Braun, Gies 2007; Gies at xqcd07 Scale fixed by α s (τ) MpL - QCD in the T,Nf plane 13

14 Just for fun Karsch Laermann Peikert Tc Nf MpL - QCD in the T,Nf plane 14

15 LATTICE STUDIES Fukugita et al. :1988 Pioneering study at N f = 10 and N f = 12 at small volumes (mainly 8 3 4) and relatively heavy masses, reporting interesting evidence for a persistence of a first order thermal phase transition at N f = 10 and discussing its sensitivity to fermion masses. Columbia: 1992 Subsequent work by the Columbia collaboration suggested a rich structure of the phase diagram, dominated by lattice artifacts and was not completely conclusive. Damgaard et al.: 1996 Consider N f = 16, the largest value of N f still compatible with asymptotic freedom, and confirm that chiral symmetry is not broken at weak coupling by observing the expected bulk transition at intermediate coupling. First attempt at a direct estimate of the beta function Iwasaki et al: 2003 A value as low as N f = 6 should be set as the lower limit of the conformal window Commonly held opinion that the strong coupling limit of QCD breaks chiral symmetry, irrespectively of N f is challenged Appelquist et al Calculation of the β function. N f = 8 is still confining in the continuum limit. N f c, the threshold of the conformal window, should be close to 12. Nice review by J. Kogut at the Lattice Strong Dynamics meeting, May2008 MpL - QCD in the T,Nf plane 15

16 THE PHASE DIAGRAM in the N f, T plane Deuzemann, Pallante, MpL 2008 NB: adjusting temperature = tuning the coupling! Nf ** Nf S cr Nf S A * Nf A 2 g(0) cr PHASES OF QCD - THE CONFORMAL WINDOW g(0) In which direction are we going? Needed control over lattice artifacts MpL - QCD in the T,Nf plane 16

17 THE ACTION Improved Kogut-Susskind fermion action, the Asqtad action which removes lattice artifacts up to O(a 2 g 2 ) one-loop Symanzik improved and tadpole improved gauge action. where with couplings defined as S = N f 4 Tr ln M(am, U, u 0) + S gauge, S gauge = i=p,r,pg β i (g 2 ) C S i Re(1 U(C)), βp β = 10/g0 2 βr = β 20u 2 ( αs) 0 βpg = β u αs, αs = 4 log u 0 / MpL - QCD in the T,Nf plane 17

18 THE SIMULATIONS Slightly modified version of the publicly available MILC code. RHMC simulations (arbitrary number of fermions) N t = 6, 12 Check of asymptotic scaling N s = 12, 20, 24 Control of Finite size effects Monte Carlo trajectories of total length τ = 0.3 to 0.4 time step δτ = to Use Blue Gene in Amsterdam MpL - QCD in the T,Nf plane 18

19 OBSERVABLES Real part of the Polyakov loop L 1 3N 3 s Re Tr x N t x 4 =1 U 4 ( x, x 4 ) Chiral Condensate a 3 ψψ = N f 4Ns 3N Tr t [ M 1], The chiral susceptibility, χ = ψψ / m = χ conn + χ disc a 2 χconn = N f [ 4Ns 3N Tr (MM) 1] t a 2 χ disc = N 2 f 16N 3 s N t [ Tr [M 1] 2 [ Tr M 1] 2], (1) MpL - QCD in the T,Nf plane 19

20 WARD IDENTITIES The susceptibilities; and χ σ χ = ψψ m = χ conn + χ disc χ π = ψψ m. are related through Ward identities (Kocic, Kogut MpL 1992) to the spacetime volume integral of the scalar (σ) and pseudoscalar (π) propagators χ σ,π = d 4 x G σ,π (x), thus implying that they should become degenerate when chiral symmetry is restored, following the degeneracy of the chiral partners. MpL - QCD in the T,Nf plane 20

21 RESULTS at N t = 6 The chiral condensate (red) and Polyakov loop (blue) as a function of the lattice coupling β. Measurements were done for three different spatial extents of the lattice N s : 12 ( ), 20 ( ) and 24 (). The critical region has been indicated by vertical lines. MpL - QCD in the T,Nf plane 21

22 The scalar χ σ (blue) and pseudoscalar χ π (red) chiral susceptibilities as a function of the lattice coupling β, confirming the degeneracy of the chiral partners in the symmetric phase. Measurements are shown for three different spatial extents of the lattice N s : 12 ( ), 20 ( ) and 24().The cumulant R π is also shown (black, top of figure). The critical region has been indicated by vertical lines. MpL - QCD in the T,Nf plane 22

23 THE CRITICAL REGION Hysteresis effect for the chiral condensate (red) and Polyakov loop (blue) within the critical region, data are for the spatial extent N s = 12. Arrows indicate the β of the configurations involved, the vertical lines indicate the estimate of the broadest critical region which covers the jump of the chiral condensate and Polyakov loop at all spatial volumes and the hysteresis cycle. MpL - QCD in the T,Nf plane 23

24 Monte Carlo history for β = 4.1 and N s = 24, at the threshold. The distribution flattens around zero in the deconfined phase, while it is randomly distributed between ± π 2 in the confined phase. The same histories are shown for runs below and above the critical region (at β = 4.0 and 4.2) in lighter colours. The points are enumerated according to their trajectory number, disregarding the first 100 trajectories after the cold start for each. MpL - QCD in the T,Nf plane 24

25 SUMMARY Nt=6 All data at N t = 6 point towards a first order phase transition. The location of the jump of the chiral condensate and the Polyakov loop at all three spatial volumes considered, and the location of the hysteresis cycle at the smallest spatial extent, allows us to give the estimate of the broadest critical region, to be 4.10 < β c < We take the occurrence of the jump of the chiral condensate and the Polyakov loop at our largest spatial volume N s = 24, as the best indicator of the critical point, giving β c (N t = 6, N s = ) = ± corresponding to the upper and lower bounds < β c (N t = 6, N s = ) < MpL - QCD in the T,Nf plane 25

26 RESULTS at N t = 12 T c = 1 a(β c )N t The chiral condensate at N t = 12 and N s = 24 in lattice units as a function of the lattice coupling β. Best fit curves are superimposed and the vertical lines indicate the critical region MpL - QCD in the T,Nf plane 26

27 Asymptotic Scaling of the Critical Temperature Results at N t = 12 and N t = 6 strongly suggest the occurrence of a first order thermal transition. To confirm this an asymptotic scaling analysis is needed in order to verify that we are actually measuring a critical temperature in the continuum real world. Given a physical temperature T c, we expect the scaling relation N t R(g c (N t )) = const. Solving for N t = 6 and N t = 12, we can predict β c (g c (N t = 6)) by knowing β c (g c (N t = 12)). By using β c (N t = 12) = 4.34±0.04 we obtain the prediction β c (N t = 6) = 4.04 ± 0.04, which deviates by less than 2% from the lattice determined β c. We also verified that a rescaling of the effective coupling β βu 4 0 improves the prediction by only 0.5% indicating that our effective coupling prescription is in fact already accounting for all the improvement MpL - QCD in the T,Nf plane 27

28 Scaling plot for the finite difference approximation to the absolute value of the first derivative of the chiral condensate as a function of T/T c, determined using the effective coupling β = 6/g 2 in the RG formula. Data at N t = 6, N s = 24, 12 (blue) are compared with data at N t = 12, N s = 24 (red). The point T/T c = 1 corresponds to β c (N t = 12) = MpL - QCD in the T,Nf plane 28

29 IN SUMMARY... β c (N f = 8, N t = 6, N s = ) = ± β c (N f = 8, N t = 6, N s = ) β c (N f = 8, N t = 12, N s = 24) = 4.34 ± β c (N f = 8, N t = 12, N s = 24) 4.38 Asymptotic scaling within 20% of the central values of the critical temperature, or, satisfied within error, consistent with the occurrence of a true thermal transition. at N f = 8, placing N f = 8 below the threshold for a conformal window. MpL - QCD in the T,Nf plane 29

30 MpL - QCD in the T,Nf plane 30

31 NEXT STEPS Precision measurements at Nf=12, Nf=14, Nf = 4 Nf c by linear extrapolation (lower bound) Nf c by direct study chiral symmetry realizations Full reconstruction of the phase diagram Critical behaviour at the chiral/conformal transition Anomalous dimensions. Large Nf vs Quenched QED4 Physics of the conformal window: spectrum of technifermions; composite Higgs. A large parameter space... challenges for thermodynamics MpL - QCD in the T,Nf plane 31

32 N f vs µ B? MpL - QCD in the T,Nf plane 32

33 Endpoint of the chiral boundary in the T, f plane? Nature of the crossover from conformal phase at low T to sqgp to QGP.. MpL - QCD in the T,Nf plane 33

34 The Phase diagram in T, µ B, N f Space? MpL - QCD in the T,Nf plane 34

35 BACKUP SLIDES MpL - QCD in the T,Nf plane 35

36 The RG equation for the running coupling can be written as ( ) q b log µ = 1 α 1 α(µ) 1 α log ( ) α (α(µ) α ) α(µ) (α α ), where α = α(q). For α, α(µ) < α we can introduce a scale defined by Λ = µ exp [ 1 b α log ( ) α α(µ) α(µ) 1 ] bα(µ), so that ( ) 1 q α = b log Λ + 1 ( ) α log. α α α MpL - QCD in the T,Nf plane 36

37 Then for q Λ the running coupling displays the usual perturbative behavior: α 1 b log ( q ), Λ For q Λ it approaches the fixed point α : α e α ( q Λ) bα. MpL - QCD in the T,Nf plane 37

38 QED / Gauged NJL Model: example of SSB at strong coupling g 1 A 0.5 S MpL - QCD in the T,Nf plane 38 alpha alpha c

39 +>)2 WD & 2# ' +.= G%4$3 / C..3 #D+=;(< 98' = >.(;$""6 MpL - QCD in the T,Nf plane 39

40 !#" %$'&)(+*, -. 0/21 3 4! "576849:1 0;+<>=!!? "@ A@".-B9:1-! C EDF1." 4"HGI!"" J K - "L K 4" KM 4 N,(,*O.L" "@ 4CP1 -Q FRS"@ 4 ">1/T"@ U-."@".DF J%$WVYX:Z4[\].H"@ +-""@ DF 45^DF1 _-! 4 "an bc d1-! e-! fbg KDF1A 9: /21 h(,*+ikjm-!1d4"@ lumn o" 4AP 4 "@ Kpf? "5o1 q 4 r"!"!1s#u-""@ DF L "@ t/u DF"vNwbx KA-! + K"q>,1z!g - f DF1 {"!S? "@ 45o" 4A 4 "@ 0"!" 1 Y"@"d1{D4DF 0 #" }G> K~{5^DF1 -! Nbc 0 S"@ 1KG]AD! q-h@ zca 4" }"1}" "E1/+ gƒ 4-!1G "@ K-! CG -! CG> C KDF -" 3"@",Dq -w {AC 4"@ G -!-wg 31z"q 41 -! 4 "1C" J- F/7"H1/c g_19: J".H-! NHbx h1 -. ˆ: K -.g 4-B C 4"@ 4 9- K]GI 4 EG> Uv9: Jƒ 4 /21 C Ke@!Š -"!1 fsgi!- h"@ E1_ K]-!1D4"@ "@ D4-9-! K N Ž MpL - QCD in the T,Nf plane 40

41 ψψ β Damgaard et al. 1996: The β dependence of the chiral order parameter for lattice sizes , 12 4 and The quark mass is amq = 0.1. The error bars are smaller than the plotting symbols. MpL - QCD in the T,Nf plane 41