From things to strings?*

Transcription

1 From things to strings?* Alberto Accardi Jlab & Hampton U. Jlab Cake Seminar, 21 Nov 2007 Based on: AA, Belli, Martellini, Zeni, PLB 431 (1998) 127, NPB 505 (1997) 540 Fucito, Martellini, Zeni, NPB 496 (1997) 259 * see 1.html

2 PART 1 Our goal: Our path: Scalar boson Non perturbative background perturbative d.o.f. (gluons) auxiliary field Intro: BRST quantization of YM theory The BFYM theory 1 First order formulation of YM (why 1st order?) A problem is found... The BFYM theory 2...and solved Gluons as a perturbation over a topological background PART 2 applications (i.e., speculations) Dynamical mechanism for confinement From things to strings: AdS/CFT hidden inside YM? 2

3 Part 1 formalism

4 BRST quantization of YM theory Classical YM lagrangian: NB: wedge products and colors always understood Local gauge symmetry: To gauge fix it, introduce: anti commuting ghosts Lagrange multiplier BRST operator (generalization of gauge symmetry): graded commutator (= anti commutator for Grassmann fields) 4

5 BRST quantization of YM theory Choose a gauge fixing condition: Define the quantum YM Lagrangian: gauge fixing fermion BRST (gauge) invariance: 5

6 The BFYM theory 1

7 1st order formulation of YM Fucito, Martellini, Zeni NPB 496 (1997) 259 Introduce auxiliary field then: ( Equivalent to YM at classical level: ) A A A B B B same function at 1 loop B A in L BFYM use B eqn. of motion Equivalent at renormalized level: only one coupling constant g A (3D) AA, Belli, Martellini, Zeni '97 (4D) AA, Belli '97 ; Fucito et al. '97 (4D) Martellini, Zeni '97 7

8 Advantages 1 (not pursued here) "Dual superconductor" mechanism for color confinement ('t Hooft '78) Wilson loop: If vacuum is a dual type II superconductor, quarks are confined: q q q R V (R) = kr T =) hw (C)i ¼ e kt R = e k R q AREA LAW Order and disorder parameter: W (C) := creates a color electric flux line along C M (C 0 ) := creates a color magnetic flux line along C' 8

9 Classification of the vacuum: phase quark confinement (dual s.c.) magnetic confinement (s.c.) full confinement Coulomb phase In YM no explicit formula for M(C) In BFYM one may use B to construct M(C): ( And ) = = BFYM = YM is confining Fucito, Martellini, Zeni (97) 9

10 Advantages 2 (today's focus) New perturbative expansion around the "topological BF theory" Definition: topological field theory "Energy momentum tensor is BRST exact: Observables such that " have metric independent v.e.v. Large class of topological invariants (e.g. Donaldson polynomials) They have no local degrees of freedom (i.e. contain no particle states) 10

11 The BF "core" of the theory (invisible in 2nd order YM) is sensitive to topology of the base manifold (i.e., manifold on which theory is defined) topology of the gauge field configurations nonlocal excitations: instantons,... YM seen as a perturbation of its BF topological core gives access to non perturbative information Loosely speaking, in BFYM: gluons seen as fluctuations over a non perturbative background 11

12 A problem, though... The "free" Lagrangian LBF has 1 more symmetry than the full LBFYM topological symmetry zero mode problem in the BFYM propagators problems in the perturbative expansion 12

13 The BFYM theory 2

14 Solution of the problem Promote the zero mode to a full field: We gain not 2 but 3 symmetries! (1) anticommuting scalars and vector ghosts The vector ghost admits a guage transformation: (1) a scalar boson!! (ghost for ghost) 14

15 Isolating non perturbative d.o.f. NOTE: I drop the subscripts (i) no local d.o.f. Moreover, e.o.m. gluons sit in correct for a local fluctuation Let's isolate the non perturbative and perturbative piece of A: non perturbative background: perturbative part = gluons: Moreover, 15

16 BRST symmetry BRST operator: s = sg + st + s' fields and symmetries of Witten's Topological YM theory (TYM) ( plus transformations of ) 16

17 Gauge fixing Gauge fixing conditions: selects instanton sector covariant gauge for background A0 background gauge for vector ghost 0 background gauge for gluons Lagrangian: A scalar boson! gluons non pert. background auxiliary field Witten's TYM no local d.o.f Perturbatively equivalent to YM (AA, Belli '97; Fucito et al. '97) 17

18 Summary of Part 1 A careful analysis of 1st order Yang Mills allows to: display a topological sector governing non perturbative dynamics interpret gluons ( ) as fluctuations over non pert. background (A0) dynamics of background is governed by the TYM lagrangian a scalar boson ( 0) naturally appears in non pert. sector Non perturbative background perturbative d.o.f. (gluons) auxiliary field A scalar boson! The new Lagrangian LBFYM is equivalent to the original LYM, but explicitly displays non perturbative d.o.f. which were hidden in the 2nd order formalism 18

19 Part 2 applications (i.e., speculations...)

20 A dynamical mechanism for color confinement and mass generation Idea: confinement is a property of the non perturbative fields A0, c0, 0, 0 consider BFYM as a perturbation of TYM

21 Step 1: freeze local d.o.f. B) = 0 When = =B=0, the BFYM reduces to the Topological Yang Mills, which is a "twisted" version of N=2 SYM: (Witten '89) Seiberg Witten analysis of N=2 SYM: (Seiberg,Witten '94) Vacua of the theory ("moduli space") are classified by 0 All vacua are degenerate (same E) 3 singularities: v = u, u, dyons magnetic monopoles + confinement as dual superconductor 21

22 Step 2: break N=2 to N=1 manually If one adds to the N=2 lagrangian terms that break SUSY to N=1 degeneracy of the vacua is lifted However, in pure N=2 SYM, the 2 supersymmetries do not allow the breaking term 1 to be generated by quantum corrections to the effective potential 1 must be added manually to lift the degeneracy 22

23 Step 3: unfreeze local d.o.f.,, B The full BFYM lagrangian has only "N=1/2" susy: the topological st corresponds via twist to the scalar combination of N=2 super charges General rule "All that is not forbidden is allowed": quantum corrections to the effective action will generate the breaking term 1 from N=2 to N=1 also a breaking term 1/2 to N=1/2 The vacua degeneracy is completely lifted only 1 vacuum survives confinement as dual superconductor dynamically generated mass scale 23

24 Step 4: check this scenario Compute the effective action Find the minimum of the quantum potential Mechanism is valid for any gauge group: why SU(3)color is confining but SU(2)xU(1) not?... if you wish to join the effort, you are welcome! 24

25 From things to strings: AdS/CFT hidden in YM?

26 The string theory connection An example of strongly interacting Yang Mills theory: N=4 Supersymmetric Yang Mills theory N=4 SYM Large Nc and strong coupling (conjectured) mathematical duality Maldacena '97 g 2 Nc! 1 Type IIB superstring on a AdS5 S5 space Weak coupling: classical gravity problem Finite temperature Black hole in AdS5 Entropy of thermal SYM matter Black hole entropy S/S0 = 3/4 viscosity of thermal SYM matter Kovtun, Son, Starinets, Policastro '02 '05 Black hole absorption of gravitons /s = 1/4 Surprise: RHIC sqgp shows /s close to 1/4 lattice QCD sqgp has S/S

27 Not only /s for N = 4 SYM Heavy quark moving in a (N=4) QGP Energy loss (drag force) Mach cones Herzog, Karch, Kovtun, Kozcaz, Yaffe; S. Gubser; Casalderrey Solana, Teaney; Fries, Gubser, Michalogiorgakis, Pufu; Gubser, Pufu, Yarom; QGP transport coefficient q^ = hkt2 i=l (average tr. mom. per unit length gained by a fast quark moving in the medium) q^sym ¼ 5 GeV2 /fm q^rhic ¼ 5 15 GeV2 /fm Liu, Rajagopal, Wiedemann 27

28 Not only /s for N = 4 SYM Heavy quarkonium dissociation does the Q Qbar bound state survive? confined Maldacena; Rey Theisen Yee; Brandhuber Itzhaki Sonnenschein Yankielowicz 3+1 dim For L<Ls: force binds Q and Qbar For L>Ls: force is screened For N=4 SYM L s = T L<Ls BH deconfined L>Ls But... wait! N=4 SYM is not QCD!! "In most circumstances, replacing QCD by N =4 super-yang-mills can be charitably described as an uncontrolled approximation" (S. Gubser) 28

29 N = 4 SYM vs. QCD at T=0 N=4 super Yang Mills: gauge fields and tree level interactions same as in QCD too many symmetries (local and global) no asymptotic freedom no confinement, no symmetry breaking no dynamical quarks, 6 scalar and 4 Weyl fermionic fields in adjoint representation One can hope that Eventually, the gravity dual of QCD is found... [see Klebanov's talk] In the QGP phase, most of the above may be irrelevant Duality applies to the non perturbative sector of QCD? [see later] 29

30 N = 4 SYM vs. QCD at finite T N=4 SYM at finite T QCD at T ~TC 3 TC conformal nearly conformal (lattice) no asymptotic freedom not an intrinsic QGP property no confinement not confined! supersymmetry is broken non susy no chiral condensate h¹ q qi = 0 no dynamical quarks, extra SUSY adjoint fields?????? Are N=4 SYM and QCD in the same universality class at (not too) high T? 30

31 Where does BFYM enter this game? N=2 SUSY is found inside pure Yang Mills at T=0: extra SUSY fields become ghosts via twist good citizens: contribute to cancel non physical d.o.f. N=2 has a confining vacuum naturally identifies gsusy = gym [see Gubser, hep th/ ] AdS/CFT duality applies to the non perturbative sector at high energy, large scales, local d.o.f. restore asymptotic freedom 1) retains a N=1/2 remnant of the supersymmetry 2) provides explicit susy breaking pattern N=2 N=1 N=1/2 BFYM as expansion over a SUSY non perturbative background gives a framework to understand the relevance of AdS/CFT computations to real life QCD, both a T=0 and T>0 [fine print: no known string dual of N=2 super Yang Mills...] 31

32 Conclusions YM can be explicitly written to display an hidden topological sector (TYM) describing non local degrees of freedom perturbatively equivalent to YM (i.e., when A0=0) a scalar boson 0 naturally appears (hidden in 2nd order YM) Similar to N=2 SYM, but with only a N=1/2 supercharge: quantum fluctuations due to local d.o.f. in BFYM may select a non trivial vacuum such that a) pure Yang Mills is confining as a dual superconductor b) v = htr Á20 i 6= 0 gives a dynamically generated mass scale Gives a framework to understand the relevance of AdS/CFT duality to real life QCD [caveat: no dual of N=2 is at present known] 32

33 Thank you!...and Happy Thanksgiving!

34 Backup slides

35 Non-central heavy-ion collisions y h x z Produced system is no more azimuthally symmetric Initial state geometry maps into final state angular distribution Nature of this mapping carries info on initial state properties 35

36 Elliptic flow 2 spatial anisotropy pressure gradients momentum anysotropy y py px x dn =N 0 [ 1 v1 pt cos 2 v2 pt cos 2... ] dpt d elliptic flow Strong scatterings local thermal equilibrium pressure gradients Self quenching effect stops on short time scales strong flow rapid equilibration 36

37 Like a perfect fluid? First time that hydro describes data without any viscosity*! Not true at SPS, not true in peripheral collisions at RHIC *viscosity = resistance of liquid to shear forces (and hence to flow) v2 pt (GeV) Hydro equations + viscosity fit data with /s ~ 0.5 (similar from lattice): 400 times less viscous than water, 10 times less than superfluid helium! Lower bound on viscosity from string theory (!!): /s > 1/4 the QGP is (nearly) a perfect fluid! 37