Lattice Study of the Conformal Window in QCD-Like Theories George - PowerPoint PPT Presentation

Lattice Study of the Conformal Window in QCD-Like Theories George Fleming Ethan Neil TA PRL 100, 171607 (2008) XIII Mexican School of Longer Paper Soon Particles and Fields 1

Beyond the Standard Model Conformal or Near-Conformal Behavior in the IR: Dynamical Electroweak Symmetry Breaking. (Walking Technicolor) New Conformal Sector? SUSY Flavor Hierarchies (Nelson & Strassler 2000/01) 2

For an asymptotically free theory, an IR fixed point can emerge already in the two-loop β function, depending on the number of fermions N f Gross and Wilczek, antiquity Caswell, 1974 Banks and Zaks, 1982 Many Others Reliable if the number of fermions is very close to the number at which asymptotic freedom is lost 3

Cartoons α * increases as N f decreases. Should be a range of N f where IR fixed point exists, not necessarily accessible in PT. (This is known in certain SUSY theories.) 4

Possibilities (1) α * < α c * (N f > N fc ) Conformal IR behavior (Non-abelian coulomb phase). (2) α * > α c * (N f < N fc ) Chiral symmetry breaking, confinement (3) α * > α c * (N f < N fc ) (fine tuning?) ~ ~ If the transition is continuous, breaking scale << Λ , ⇒ Walking at intermediate scales. 5

Questions 1. Value of N fc ? 2. Order of the phase transition? 3. Physical states below and near the transition? 4. Implications for EW precision studies? (The S parameter etc)? 5. Implications for the LHC? 6

N fc in SU(N) QCD • Degree-of-Freedom Inequality (Cohen, Schmaltz, TA 1999). Fundamental rep: N fc ≤ 4 N[1 – 1/18N ² + …] N fc ≅ 4 N • Gap-Equation Studies, Instantons (1996): • Lattice Simulation (Iwasaki et al, Phys Rev D69, 014507 2004): 6 < N fc < 7 For N = 3 7

N fc in SUSY SU(N) QCD Degree of Freedom Inequality: N fc ≤ (3/2) N Seiberg Duality: N fc = (3/2) N !! Weakly coupled magnetic dual in the vicinity of this value 8

Some Quasi-Perturbative Studies of the Conformal Window in QCD-like Theories 1. Gap – Equation studies in the mid 1990s 2. V. Miransky and K. Yamawaki hep-th/9611142 (1996) 3. E. Gardi, G. Grunberg, M. Karliner hep-ph/9806462 (1998) 4. E. Gardi and G. Grunberg “The IRFP is perturbative in the JHEP/004A/1298 (2004) entire conformal window ” 5. Kurachi and Shrock, hep-ph/0605290 6. H. Terao and A. Tsuchiya arXiv:0704.3659 [hep-ph] (2007) 9

Lattice-Simulation Study of the Extent of the Conformal window in an SU(3) Gauge Theory with Dirac Fermions in the Fundamental Representation 10

Previous Lattice Work with Many Light Fermions 1. Brown et al (Columbia group) Phys. Rev. D12, 5655 (1992) = 8 N f 2. Damgaard, Heller, Krasnitz and Oleson, hep-lat/9701008 = 16 N f ( ) 3. R. Mahwinney, hep/lat/9701030(1) , → 4 N f Nucl.Phys.Proc.Suppl.83:57-66,2000. e-Print: hep-lat/0001032 4. C. Sui, Flavor dependence of quantum chromodynamics. PhD thesis, Columbia University, New York, NY, 2001. UMI-99-98219 5. Iwasaki et al, Phys. Rev, D69, 014507 (2004) 11

Focus:Gauge Invariant and Non- Perturbative Definition of the Running Coupling Deriving from the Schroedinger Functional of the Gauge Theory ALPHA Collaboration: Luscher, Sommer, Weisz, Wolff, Bode, Heitger, Simma, … 12

Using Staggered Fermions as in U. Heller, Nucl. Phys. B504, 435 (1997) Miyazaki & Kikukawa O(a 2 ) Chiral Breaking Remaining Continuous Chiral Symmetry Focus on N f = Multiples of 4: 16: Perturbative IRFP 12: IRFP “expected”, Simulate 8 : IRFP uncertain , Simulate 4 : Confinement, ChSB 13

The Shroedinger Functional • Transition amplitude from a prescribed state at t=0 to one at t=T (Dirichlet BC). • Euclidean path integral with Dirichlet BC in time and periodic in space (L) to describe a constant chromo-electric background field. [ ] ′ ′ ′ ζ ζ ζ ζ = , , ; , , Z W W [ ] ∫ ′ ′ ′ ′ − − ζ ζ ζ ζ χ χ ( , ) ( , , , , , ) S W W S W W DUD D e G F 14

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Abelian Boundary Fields ( ) ( ) / / / = i o 1 L i o 2 L i o 3 L , , , W x diag e e e k ( ) ( ) / / / ′ = i o L i o L i o 3 L 1 2 , , . W x diag e e e k π π = − + η = − η = − + η / , / , / , 1 1 o o o 1 2 3 2 2 3 3 π π 2 ′ ′ ′ = − π − η = + η = + η / , / 1 , / 1 . o o o 1 2 3 2 2 3 3 1 • Constant chromoelectric background field of strength L • Can set = 0 m f 16

Schroedinger Functional (SF) Running Coupling on Lattice Define: − ∂ 1 1 ≡ log , Z ( ) ∂ η η = 2 , 0 g L T k ( ) ... 1 ( ) 2 = + + + 0 1 0 g 0 2 g 0 Response of system to small changes in the background field. 2 ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ π π 2 2 ⎛ ⎞ 2 L a a ⎜ ⎟ ⎜ ⎟ = + 12 ⎜ ⎟ sin sin ⎢ ⎥ k ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 3 3 ⎝ ⎠ ⎝ ⎠ a ⎣ LT LT ⎦ 17

SF Running Coupling Then, to remove the O(a) bulk lattice artifact ⎡ ⎤ 1 1 1 1 = + ⎢ ) ⎥ ( ) ( ) ( − + 2 2 2 2 , , ⎣ ⎦ g L g L L a g L L a Depends on only one scale L Look for conformal symmetry (IRFP) at the box scale L 18

Loop Expansion ∂ ( ) ( ) ( ) ( ) ( ) .... = β = + + + 2 2 4 6 8 ( ) L g L g L b g L b g L b g L 1 2 3 ∂ L ⎛ ⎞ ⎛ ⎞ 2 2 2 38 = − = − ⎜ ⎟ ⎜ ⎟ 11 , 102 b N b N ( ) ( ) 1 2 π 2 π 4 ⎝ 3 f ⎠ ⎝ 3 f ⎠ 4 4 ( ) − b c b c c = + − 2 2 1 3 2 M S b b 3 3 π π 2 2 2 8 ⎡ ⎤ 1 2857 5033 325 = − + 2 M S b N N ⎢ ⎥ ( ) 3 6 π f ⎣ 2 18 54 ⎦ 4 f = + 1 . 256 0 . 04 c N 2 f = + + − 2 2 1 . 20 0 . 14 0 . 03 c c N N 3 2 19 f f

Loop Expansion ⎛ ⎞ 2 * 2 = g ⎜ ⎟ ≈ 01 . = 0 . 47 IRFP at 16 ⎜ ⎟ g N π 2 4 SF ⎝ ⎠ f ⎛ ⎞ 2 * 2 = g ⎜ ⎟ ≈ 13 = 5 . 18 IRFP at . 12 g ⎜ ⎟ N π 2 4 SF f ⎝ ⎠ ≤ 8 No perturbative IRFP N f 20

Loop Expansion β = Linearize near the 12 N f IRFP [ ] β 2 ≅ γ SF − * 2 2 ( ( )) ( ) g L g g L Then: ( ) const → ∞ 2 ⎯ ⎯ → ⎯ * 2 − L g L g γ SF L 21

Lattice Simulations MILC Code (Heller) = 8 , 12 N Staggered Fermions f Range of Lattice Couplings g 0 ² (= 6/ β ) and Lattice Sizes L/a → 20 O(a) Lattice Artifacts due to Dirichlet Boundary Conditions 22

Statistical and Systematic Error 1. Numerical-simulation error 2. Interpolating-function error 3. Continuum-extrapolation error Statistics Dominates 23

24 Data 8 = f N

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28 Data with Fits 8 = f N

Renormalization Group (Step Scaling) ⎛ ⎞ ⎛ ⎞ ( ) a L a ⎜ ⎟ = 2 2 2 2 ⎜ ⎟ , , , g g g g L ⎜ ⎟ 0 0 ⎝ ⎠ ⎝ ⎠ L L L 0 0 ( ) ⎯ → ⎯ → ⎯ 0 2 1 ln a g L a 0 0 ⎛ ⎞ ⎛ ⎞ ( ) L L ⎛ ⎞ → ⎜ ⎟ ⎜ ⎟ ⎯ ⎯ ⎯ ⎯ → ≡ L 0 2 2 2 , a L ⎜ ⎟ = 2 0 g g L g ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎝ ⎠ ⎝ ⎠ L L ⎝ ⎠ L 0 0 0 ′ ⎛ ′ ⎛ ⎞ ⎞ a → ( ) L a L 0 ′ ⎛ ⎞ ⎯ ⎯ → ⎯ 2 2 2 L ⎜ ⎟ ⎜ ⎟ , , = L ⎜ 2 ⎟ g g L g ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ L L L L 29

N f =8 Extrapolation Curve 30

N f = 8 Continuum Running 31

N f = 8 Features 1. No evidence for IRFP or even inflection point up through ( ) ≈ 2 15 . g L ( ) ∗ π α c ≈ 1 4 2. Exceeds rough estimate of strength required to break chiral symmetry, and therefore produce confinement. Must be confirmed by direct lattice simulations. 3. Rate of growth exceeds 3 loop perturbation theory. 4. Behavior similar to quenched theory [ALPHA N.P. Proc. Suppl. 106, 859 (2002)] and N f =2 theory [ALPHA, N.P. B713, 378 (2005)], but slower growth as expected. 32

33 Data 12 = f N

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39 Data with Fits 12 = f N

N f =12 Extrapolation Curve 40

N f = 12 Continuum Running 41

Conclusions 1. First lattice evidence that for an SU(3) gauge theory with N f Dirac fermions in the fundamental representation 8 < N fc < 12 2. N f =12: Relatively weak IRFP 3. N f =8: Confinement and chiral symmetry breaking – in disagreement with Iwasaki et al Employing the Schroedinger functional running coupling defined at the box boundary L 42

Things to Do 1. Refine the simulations at N f = 8 and 12 and examine other values such as N f =10. 2. Study the phase transition as a function of N f. 3. Consider other gauge groups and representation assignments for the fermions 4. Examine physical quantities such as the static potential (Wilson loop) and correlation functions. 43