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Gradient flow running coupling: SU(2) with 6 fundamental flavors - - PowerPoint PPT Presentation

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Gradient flow running coupling: SU(2) with 6 fundamental flavors Viljami Leino Kari Rummukainen Joni Suorsa Kimmo Tuominen University of Helsinki and Helsinki Institute of Physics 28.07.2016 Lattice 2016, Southampton Motivation Nearly

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SLIDE 1

Gradient flow running coupling: SU(2) with 6 fundamental flavors

Viljami Leino Kari Rummukainen Joni Suorsa Kimmo Tuominen

University of Helsinki and Helsinki Institute of Physics 28.07.2016 Lattice 2016, Southampton

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SLIDE 2

Motivation

  • Nearly

conformal theories can have walking behavior needed by technicolor

  • SU(2) with 8 massless fer-

mions has a fixed point 1

1 2 3 4 5 6 7 8 9

g2

GF 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08

σ(g2

GF, 2)/g2 GF

continuum 16-32 2-loop 4-loop MS

  • Previous studies at Nf =6

inconclusive 2 3 4

  • 4-loop MS IRFP g2 ∼ 30

4 8 12 1 2 3 4 5 6 g

  • 0.5
  • 0.25

0.25 0.5 β

1 / 14

1 V. Leino et al. Lattice 2015 (hep-lat/1511.03563 )

,

2 T. Karavirta et al. JHEP 1205 (2012) 003 (hep-lat/1111.4104) , 3 T. Appelquist et al. Phys. Rev. Lett. 112, 111601 (2014) (hep-lat/1311.4889) 4 M. Hayakawa et al. Phys. Rev. D 88, 094504 (2013) (hep-lat/1307.6997)

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SLIDE 3

Model

  • HEX-smeared1 Wilson-clover action
  • Schrödinger functional
  • Use trivial (Dirichlet) boundaries (no background field)
  • Used to reach zero mass (Tune the κcr at L = 24)
  • Allows the measurement of mass anomalous dimension
  • Lattice sizes: 8,12,16,18,20,24,30,(36)
  • Use step scaling step s = 3/2 ( 8-12, 12-18, 16-24, 20-30)
  • Can compare to s = 2 at 8-16 and 12-24
  • β between 8 and 0.5
  • We run into bulk phase transition at β < 0.5
  • Smaller lattices ∼ 80 000 trajectories, larger ∼ 15 000

2 / 14

1 S. Capitani, S. Durr and C. Hoelbling, JHEP 0611 (2006) 028

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SLIDE 4

Gradient Flow

  • Use the gradient flow 1 2

g2

GF = t2

N E(t + τ0a2)

  • Flow can be evolved using both Wilson plaquette (W) and

Lüscher-Weisz (LW) actions

  • Energy can be measured with both clover and plaquette

definitions

  • We use LW and clover unless otherwise specified
  • Fix flow time t to L by setting scale: c =

√ 8t/L = 0.3

  • Use τ0 correction to tune down the a2 effects 3
  • Measuring also the topological charge:

Q = 1 32π2

  • x

ǫµναβG a

µν(x; t)G a αβ(x; t) 3 / 14

1 M. Luscher and P. Weisz , JHEP 1102 (2011) 051 (hep-th/1101.0963)

,

2 P. Fritzsch and A. Ramos , JHEP 1310 (2013) 008 (hep-lat/1301.4388) , 3 A. Cheng, A. Hasenfratz, Y. Liu, G. Petropoulos and D. Schaich. JHEP 1405 (2014) 137 (hep-lat/1404.0984)

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SLIDE 5

Measured couplings

8 12 16 20 24 28 32 36 L/a 2 4 6 8 10 12 14 16 18 20 22 24 26 g

2

β=8 β=6 β=5 β=4 β=3 β=2 β=1.7 β=1.5 β=1.3 β=1.1 β=1 β=0.9 β=0.8 β=0.75 β=0.7 β=0.65 β=0.6 β=0.55 β=0.53 β=0.5

4 / 14

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SLIDE 6

Topology

  • 0.001
  • 0.0005

0.0005 0.001 0.0015 200 400 600 800 1000 1200 1400 β=2

  • 1
  • 0.5

0.5 1 1.5 β=0.7

  • 10
  • 5

5 10 15 β=0.53

L-W evolved flow

  • 0.001
  • 0.0005

0.0005 0.001 0.0015 200 400 600 800 1000 1200 1400 β=2

  • 1
  • 0.5

0.5 1 1.5 β=0.7

  • 10
  • 5

5 10 15 β=0.53

Wilson evolved flow

  • Topology frozen at small couplings, becomes unfrozen at

largest couplings

  • LW evolved flow fluctuates more
  • Don’t use configurations that are frozen to nonzero values
  • Projecting δQ,0 1 could work, but for Nf = 8 effects were small

5 / 14

1 P. Fritzsch et al. PoS Lattice 2013, 461 (2014) (hep-lat/1311.7304)

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SLIDE 7

Step scaling function

  • Interpolate couplings using a rational function, m = 7, n = 2

g2

GF(g2 0 , L/a, t) = g2

1 + m

i=1 aig2i

1 + n

j=1 bjg2j

.

  • Estimate systematic errors by changing the fit parameters
  • Step scaling function:

Σ(u, s, a/L) = g2

GF(g0, s L

a)

  • g2

GF (g0, L a )=u

, σ(u, s) = lim

a/L→0 Σ(u, s, a/L)

  • Extrapolate to continuum limit:

Σ(u, s, a/L) = σ(u, s) + c(u)(L a)−2

  • Fix τ0 to minimize a2 effects

6 / 14

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SLIDE 8

Fixing τ0

0.005 0.01 0.015 0.02

(a/L)2

4.8 5.0 5.2 5.4 5.6 5.8 6.0

σ(g2, 2) τ0 = 0 τ0 = 0. 05 τ0 = 0. 1

u = 5

0.005 0.01 0.015 0.02

(a/L)2

1 2 3 4 5 6 7 8 9 10 11 12 13 14

σ(g2, 2)

τoptimal vs. τ0 = 0

  • Drop the smallest lattice from continuum extrapolation
  • Estimate: τoptimal = 0.012 log(1 + 20g2) (Preliminary)
  • Logarithm makes sure the τ0 doesn’t grow too large

7 / 14

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SLIDE 9

Different discretizations c = 0.3 , τ0 = 0

0.005 0.01 0.015 0.02

(a/L)2

1.0 1.2 1.4 1.6 1.8 2.0 2.2

σ(g2, 2) W Clover W Plaq LW Clover LW Plaq

u = 2

0.005 0.01 0.015 0.02

(a/L)2

5 6 7 8 9 10 11 12 13

σ(g2, 2) W Clover W Plaq LW Clover LW Plaq

u = 11

  • Plaquette and Clover agree on continuum limit, plaquette has

stronger discretization effects

  • LW and W diverge slightly on large couplings, W has stronger

discretization effects

8 / 14

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SLIDE 10

Step scaling on the lattice c = 0.3

2 4 6 8 10 12 14 16 18 20 22 24 g

2

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 σ(g

2,2)/g 2

L=8 L=12 L=16 L=20 L=24 2-loop 4-loop 3-loop

s = 3/2

2 4 6 8 10 12 14 16 18 20 22 24 g

2

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 σ(g

2,2)/g 2

L=8 L=12 L=18 2-loop 4-loop 3-loop

s = 2

9 / 14

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SLIDE 11

Continuum limit

5 10 15 20

g2

GF 0.90 0.95 1.00 1.05 1.10 1.15

σ(g2

GF, 2)/g2 GF

continuum 20-30.0 2-loop 3-loop MS 4-loop MS

12 − 30

5 10 15 20

g2

GF 0.90 0.95 1.00 1.05 1.10 1.15

σ(g2

GF, 2)/g2 GF

continuum 16-24.0 2-loop 3-loop MS 4-loop MS

8 − 24

  • L20-30 has less statistics than L8-12
  • L8-12 behaves oddly on strong coupling and L8 was not used

when defining τ0

10 / 14

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SLIDE 12

Effects of parameters

5 10 15 20

g2

GF 0.90 0.95 1.00 1.05 1.10 1.15

σ(g2

GF, 2)/g2 GF

LW W 2-loop 3-loop MS 4-loop MS

5 10 15 20

g2

GF 0.90 0.95 1.00 1.05 1.10 1.15

σ(g2

GF, 2)/g2 GF

Clover Plaq 2-loop 3-loop MS 4-loop MS

5 10 15 20

g2

GF 0.90 0.95 1.00 1.05 1.10 1.15

σ(g2

GF, 2)/g2 GF

τ0 No τ0 2-loop 3-loop MS 4-loop MS

5 10 15 20

g2

GF 0.90 0.95 1.00 1.05 1.10 1.15

σ(g2

GF, 2)/g2 GF

continuum c=0.4 20-30.0 2-loop 3-loop MS 4-loop MS

11 / 14

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SLIDE 13

Mass anomalous dimension

  • Schrödinger functional pseudoscalar density renormalization

constant allows calculation of γ 1

  • Interpolate ZP with ZP = 1 + 5

i=1 aig2i

  • Near fixed point approximate as γ∗

ZP(g0, L a) = √Nf1 fp( 1

2 L a) ,

ΣP(u, a L) = ZP(g0, sL

a )

ZP(g0, L

a)

  • g2

GF =u

σP(g2) = lim

a→0 ΣP(g2, a

L) , γ∗ = −log σP(g2) log s

12 / 14

1 S. Capitani, M. Luscher, R. Sommer and H. Witting Nucl. Phys. B 544 (1999) (hep-lat/9810063)

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SLIDE 14

γ∗

2 4 6 8 10 12 14 16 18 20 22 24 g

2

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 γ L=8 L=12 L=16 L=20 2 4 6 8 10

g2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

ZP L=30

1 2 3 4 5 6 7 8 9 10 11 12

g2

GF 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

γ ∗ continuum 20-30 Perturbative

1 2 3 4 5 6 7 8 9 10 11 12

g2

GF 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

γ ∗ continuum 20-30 Perturbative

13 / 14

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SLIDE 15

Conclusions

  • Finite volume GF step scaling works at strong coupling
  • This choice of action, boundaries and smearing allows us to

reach relatively small β before running into a bulk phase transition

  • Topological freezing mostly problem only on certain range of

β’s

  • Nf = 6 seems to approach a IRFP around g2 ∼ 15
  • Check also the posters:
  • γ with spectral density method – Joni Suorsa
  • Spectrum of Nf = 2, 4, 6, 8 – Sara Tähtinen

14 / 14