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

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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

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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)

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For an asymptotically free theory, an IR fixed point can emerge already in the two-loop β function, depending on the number of fermions Nf

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

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Cartoons

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

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Possibilities

(1) α* < αc* (Nf > Nfc)

Conformal IR behavior (Non-abelian coulomb phase).

(2) α* > αc* (Nf < Nfc)

Chiral symmetry breaking, confinement

(3) α* > αc* (Nf < Nfc) (fine tuning?)

If the transition is continuous, breaking scale << Λ , ⇒ Walking at intermediate scales.

~ ~

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Questions

• 1. Value of Nfc?
• 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?

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Nfc in SU(N) QCD

• Degree-of-Freedom Inequality (Cohen, Schmaltz, TA 1999).

Fundamental rep: Nfc ≤ 4 N[1 – 1/18N² + …]

• Gap-Equation Studies, Instantons (1996):

Nfc ≅ 4 N

• Lattice Simulation (Iwasaki et al, Phys Rev D69, 014507 2004):

6 < Nfc < 7 For N = 3

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Nfc in SUSY SU(N) QCD

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

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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)

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

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Previous Lattice Work with Many Light Fermions

1. Brown et al (Columbia group) Phys. Rev. D12, 5655 (1992) 2. Damgaard, Heller, Krasnitz and Oleson, hep-lat/9701008 3.

• R. Mahwinney, hep/lat/9701030(1) ,

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)

8 =

f

N

16 =

f

N

( )

4 →

f

N

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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, …

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Using Staggered Fermions as in

• U. Heller, Nucl. Phys. B504, 435 (1997)

Miyazaki & Kikukawa

O(a2) Chiral Breaking Remaining Continuous Chiral Symmetry

Focus on Nf = Multiples of 4: 16: Perturbative IRFP 12: IRFP “expected”, Simulate 8 : IRFP uncertain , Simulate 4 : Confinement, ChSB

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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

χ χ ζ ζ ζ ζ

′ ′ ′ − ′ −

∫

= ′ ′ ′

W W S W W S

F G

e D DUD W W Z

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Picture

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Abelian Boundary Fields

• Constant chromoelectric background field of strength
• Can set

( )

( )

( )

( )

. , , , , ,

3 2 1 3 2 1 L

• i

L

• i

L

• i

L

• i

L

• i

L

• i

e e e diag x W e e e diag x W

k k

/ / / / / /

= ′ =

. 3 2 , 3 , , 3 , , 3

2 1 3 2 1 2 1 2 1 3 2 1 2 1

η π η π η π η π η η π + = ′ / + = ′ / − − = ′ / + − = / − = / + − = /

• L

1

=

f

m

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Schroedinger Functional (SF) Running Coupling on Lattice

Define:

Response of system to small changes in the background field.

( )

, log 1 , 1

2 =

∂ ∂ − ≡

η

η Z k T L g

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = LT a LT a a L k 3 sin 3 2 sin 12

2 2 2

π π

( )

( ) ...

1 1

2 2

+ + + = g g

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SF Running Coupling

Then, to remove the O(a) bulk lattice artifact Depends on only one scale L Look for conformal symmetry (IRFP) at the box scale L

( ) ( ) ( )⎥

⎦ ⎤ ⎢ ⎣ ⎡ + + − = a L L g a L L g L g , 1 , 1 2 1 1

2 2 2

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Loop Expansion

( )

( )

( ) ( ) ( ) ....

) (

8 3 6 2 4 1 2 2

+ + + = = ∂ ∂ L g b L g b L g b L g L g L L β ( ) ( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − =

f f

N b N b 3 38 102 4 2 , 3 2 11 4 2

4 2 2 1

π π

( )

2 2 3 1 2 2 2 3 3

8 2 π π c c b c b b b

S M

− − + =

( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − =

2 6 3

54 325 18 5033 2 2857 4 1

f

N N b

f S M

π

f

N c 04 . 256 . 1

2

+ =

2 2 2 3

03 . 14 . 20 . 1

f f

N N c c − + + =

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Loop Expansion

IRFP at IRFP at No perturbative IRFP

16 =

f

N

47 . *2 =

SF

g ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≈ 01 . 4

2 2

π g

12 =

f

N

18 . 5 *2 =

SF

g

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≈ 13 . 4

2 2

π g

8 ≤

f

N

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Loop Expansion

Linearize near the IRFP Then:

[ ]

) ( )) ( (

2 2 * 2

L g g L g

SF −

≅ γ β

( )

γ L const g L g

SF L

− ⎯ ⎯ → ⎯

∞ → 2 * 2

β

12 =

f

N

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Lattice Simulations

MILC Code (Heller) Staggered Fermions Range of Lattice Couplings g0² (= 6/β ) and Lattice Sizes L/a → 20 O(a) Lattice Artifacts due to Dirichlet Boundary Conditions

12 , 8 =

f

N

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Statistical and Systematic Error

• 1. Numerical-simulation error
• 2. Interpolating-function error
• 3. Continuum-extrapolation error

Statistics Dominates

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Data

8 =

f

N

25

26

27

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Data with Fits

8 =

f

N

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Renormalization Group (Step Scaling)

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

2 2 2 2

, , , L a L L L g g L a g g

( )

a L g

a 2

ln 1 ⎯ ⎯ → ⎯ →

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎯ ⎯ ⎯ → ⎯

→ 2 2 2

, L L g L L L g g

L a

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ ⎯ ⎯ → ⎯ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′

→

L L g L a L L L g g

L a 2 2 2

, ,

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 L L

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ′ 2 L L

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Nf=8 Extrapolation Curve

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Nf = 8 Continuum Running

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Nf = 8 Features

1. No evidence for IRFP or even inflection point up through . 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 Nf=2 theory [ALPHA, N.P. B713, 378 (2005)], but slower growth as expected.

( )

15

2

≈ L g

( )

4 1 ≈

∗ π

αc

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Data

12 =

f

N

34

35

36

37

38

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Data with Fits

12 =

f

N

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Nf=12 Extrapolation Curve

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Nf = 12 Continuum Running

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Conclusions

1. First lattice evidence that for an SU(3) gauge theory with Nf Dirac fermions in the fundamental representation 8 < Nfc < 12

• 2. Nf=12: Relatively weak IRFP
• 3. Nf=8: Confinement and chiral symmetry

breaking – in disagreement with Iwasaki et al Employing the Schroedinger functional running coupling defined at the box boundary L

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Things to Do

1. Refine the simulations at Nf = 8 and 12 and examine other values such as Nf =10.

• 2. Study the phase transition as a function of Nf.

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.

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• 5. Examine chiral symmetry breaking directly:

<ψ ψ> at zero temperature

• 6. Apply to BSM Physics. Is S naturally small as Nf

→Nfc due to approximate parity doubling? Includes the contribution of the [ - 1 - 3 ] pseudo- Nambu-Goldstone bosons present in the model.

( )

( ) ( ) [ ]

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − Π − Π = ∫

∞ 2 , 3 , ,

1 1 48 1 Im Im 4

ref H ref H AA VV ref H

m s s m s s s ds m S θ π

2

f

N

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LSD Collaboration Lattice Strong Dynamics

• J. C. Osborn

Argonne National Laboratory

• R. Babich, R. C. Brower, M. A. Clark, C. Rebbi, D.

Schaich Boston University

• M. Cheng, T. Luu, R. Soltz, P. M. Vranas

Lawrence Livermore National Laboratory

• T. Appelquist, G. T. Fleming, E. T. Neil

Yale University