Luigi Del Debbio - Swansea Universitypyweb.swan.ac.uk/~pybl/LargeNTalks/Deldebbio.pdfFlavor physics • ETC: mass term for the SM fermions, FCNC −→ ∆L ∝ 1 M2 ETC hΨΨ¯ iETC

Walking dynami s on the latti eLuigi Del Debbio

[email protected]

University of Edinburgh

walking dynamics, swansea 07/09 – p.1/47

Acknowledgements

work in collaboration with:

bursa, keegan, kerrane, lucini, moraitis, patella, pica, pickup, rago


Simulations beyond QCD

• simulations with light, dynamical fermions w/out compromising the robustness of theformulation

• dynamics different from QCD + we don’t know the answer!!

• lattice simulations: robust quantitative tool for NP QFT

SU(N) gauge theories with fermions in higher–dimensional reps

• nonperturbative physics at LHC: candidate theory for technicolor

• framework and implementation

• first numerical results [catterall et al 07/08, ldd et al 08, rummukainen et al 08/09, degrand et al 08,

appelquist et al 07/09, deuzeman et al 08/09]


Flavor physics

• ETC: mass term for the SM fermions, FCNC

−→ ∆L ∝1

M2ETC

〈ΨΨ〉ETC ψψ,1

M2ETC

ψψ ψψ

• running of 〈ΨΨ〉: (near) conformal behaviour

〈ΨΨ〉ETC = 〈ΨΨ〉TC exp

Z METC

ΛTC

dµ

µγ(µ)

!


Conformal IR behaviour

β(g) = µd

dµg(µ) = −β0g

3−β1g5+O(g7)

β0 =1

(4π)2

»

11

3C2(A) −

4

3TRnf

–

β1 =1

(4π)4

»

34

3C2(A)2 −

20

3C2(A)TRnf

−4C2(R)TRnf

˜

→ nf : number of light fermions

(4π)2β0 (4π)4β1

SU(3), nf = 3 fund 9 64

SU(2), nf = 2 adj 2 -40

0.0 0.5 1.0 1.5 2.0 2.5 3.0g

-2.00

-1.00

0.00

1.00

β(g)

1-loop2-loop

0.0 0.5 1.0 1.5 2.0 2.5 3.0g

-0.06

-0.04

-0.02

0.00

0.02

β(g)

1-loop2-loop


Walking theories - scheme dependence

• power-enhancement by near–conformalbehaviour

• IRFP is scheme-independent

β(g) = µd

dµg(µ)

change of scheme: g′ = Φ(g), Φ′(g) > 0

β′(g′) = Φ′(g)β(g)

universality of the first two coefficients

• β-function away from IRFP is notuniversal!

• m′ = mF (g2)

γ′ = γ + βF ′

F

0.0 1.0 2.0 3.0g

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

β(g)

QCD-likeIRFPwalking?IRFP


Change of scheme

0.5 1.0 1.5 2.0 2.5 3.0 3.5g

-10

-8

-6

-4

-2

ΒHgL

1.0 1.5 2.0 2.5 3.0 3.5g

0.8

1.0

1.2

1.4

g'


DEWSB @ LHC

• collider phenomenology via effective theories

• effective lagrangian for the low–lying statesLHC observables in terms of the LEC [giudice et al, sannino et al, lane et al]

• concrete proposals for underlying TC theories can be tested on the latticee.g. Minimal Walking TC [sannino et al 05]

→ only two free parameters

• lattice simulations provide first principle calculations→ phase structure of the theory, spectrum, LEC

• light dynamical fermions play a key role for non QCD–like behaviour

• understand the effect of systematic errors in lattice results


Phase diagram [sannino 08]


Perturbation theory

[ldd, frandsen, panagopoulos, sannino 08]

Perturbative computations can be easily generalized:

• vertices with fermions and gluons involve generators TaR

• vertices with 3 and 4 gluons involve generators TaF

• at one–loop replace Casimir operators as necessary

• numerical integrals are independent of the color structure

amc(g0) = g20Σ(1) + g40Σ(2)

Σ(1) = 2C2(R) [c1 + c2]

Σ(2) = C2(R)Nd1 + 2C2(R)TRnfd2 + C2(R)C2(F )d3 + C2(r)2d4

ZA = 1 −g20

16π2C2(R) × 15.7

ΛMS/Λlat

˛

˛

SU(3),nf =2,2S= 1469.59

ΛMS/Λlat

˛

˛

SU(2),nf =2,2S= 6884.36


Chiral symmetry

For a generic N 2–index representations are complex:

SU(nf ) × SU(nf ) −→ SU(nf )

=⇒ usual pattern of chiral symmetry breaking

For the adjoint representation

chiral symmetry breaking pattern for nf :

SU(2nf ) −→ SO(2nf )

Goldstone bosons are coupled to EW gauge bosons:

FPS = v ≈ 250 GeV

→ can be used to set the scale


RG flows in QCD

• the theory has two relevantparameters g, m

• renormalized trajectories lie in the(g,m) plane – perfect actions

• simulations are performed away fromthis plane

• aµ small in order to have small dis-cretization effects

am

g

g’


Conformal fixed point?

• scale invariance is broken by m AND1/L

• large physical volume + light masses!

• deviations from QCD spectrum

• Schrödinger functional/twisted BC al-low m = 0

am

g

g’


Conformal fixed point?

g’

am

Aoki

small e.v.

g


Conformal spectrum

QCD-like conformal

mPS

Λhad

mPS

ΛYM

[e.g. miranski 95]


Higher representations

Wilson–Dirac operator:

D =1

2

X

µ

ˆ

γµ

`

∇µ + ∇∗µ

´

− a∇∗µ∇µ

˜

+m,

i.e.

X

y

D(x, y)ψ(y) = −1

2a

n

X

µ

h

(1 − γµ)UR(x, µ)ψ(x+ µ)+

(1 + γµ)UR(x− µ, µ)†ψ(x− µ)i

− (8 + 2am)ψ(x)o

,

κ = 1/(8 + 2am)

Link variables:

U(x, µ) = exph

i Aaµ(x)Ta

f

i

, N ×N matrix

UR(x, µ) = exph

i Aaµ(x, µ)Ta

R

i

, dR × dR matrix


HMC for higher representations

[ldd, patella, pica 08]

Including the fermionic determinant:

Z =

Z

DUDφDφ∗ exp[−SG −‚

‚Q−1φ‚

‚

2]

=

Z

DΠDUDφDφ∗ exp[−1

2(Π,Π) − SG −

‚

‚Q−1φ‚

‚

2], Q = γ5D

Molecular dynamics evolution:

d

dtΠ(x, µ) = −

δH

δU(x, µ)=X

k

Fk(x, µ)

d

dtU(x, µ) = Π(x, µ)U(x, µ)

performed numerically through discrete leapfrog integration w. time-step dτTotal trajectory length τ = Ndτ – cost ∝ N〈Niter〉

performance of the algorithm for light fermion masses is crucial


Exploring the chiral regime of MWT

[edinburgh-swansea 09]

-1.5 -1.4 -1.3 -1.2 -1.1 -1(am0)

0.42

0.44

0.46

0.48

0.5

g 02 /4

critical line [Catterall et al 08]pert theory [ldd et al 08]


Taming systematic errors

In order to keep systematics under control:

a/L≪ ma≪“ r0

a

”−1≪ 1

In QCD: L/a ∼ 48, r−10 ≃ 400 MeV, mπ ≃ 200 MeV, a−1 ≃ 2 GeV

0.02 ≪ 0.05 ≪ 0.2 ≪ 1

if IRFP is present: conformal symmetry is broken by the mass

• same conditions apply on L,m, and a

• more delicate to define a scale like r0• large volumes and small masses are mandatory


Stability of the HMC

[ldd, giusti, luscher, petronzio, tantalo 05]

0

0.1

0.2

p(µ)

0 5 10 15 20 25 30µ [MeV]

0

0.1

0.2

• integration instabilities / reversibility

• ergodicity problem: HMC stuck in asector with η 6= 0

• sampling of observables: p(µ)/µ2

difficult regime for simulations:

m→ 0

a, V fixed

→ check that runs are in a safe regime!!


Numerical results for SU(2) adj

[ldd, patella, pica 08]

0

0.1

0.2

κ=0.18587

0

0.1

0.2

κ=0.18657

0

0.1

0.2

κ=0.18691

0

0.1

0.2

κ=0.18727

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2aλ

0

0.1

0.2

κ=0.18797

data from 16 × 83 lattice, β = 2.0


Numerical results for SU(2) adj

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2plaq_32x16x16x16b2.25m1.175.out

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2plaq_32x16x16x16b2.25m1.18.out

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

0.3plaq_32x16x16x16b2.25m1.185.out

0 0.02 0.04 0.06 0.08 0.10

0.01

0.02

0.03

0.04

0.05plaq_32x16x16x16b2.25m1.19.out

data from 32 × 163 lattice, β = 2.25


Chiral limit (1)

-1.25 -1.20 -1.15 -1.10 -1.05 -1.00 -0.95a m0

0.00

0.10

0.20

0.30

0.40

a m

16x83

24x123

64x243

32x163

β = 2.25

a mc = -1.202(1)

data from β = 2.25 lattices


Chiral limit (2)

-1.20 -1.15 -1.10a m0

0.00

0.10

0.20a

m

16x83

24x123

64x243

32x163

β = 2.25

a mc = -1.202(1)


Comparison with other studies

0.12 0.15 0.18 0.21 0.24κ

0

0.3

0.6

0.9

1.2

mQ a

βL=1.3

βL=1.7

βL=1.9

βL=2.2

βL=2.5

SU(2)-adjoint

[rummukainen et al 08]


Pseudoscalar mass (1)

0.00 0.10 0.20 0.30 0.40a m

0.0

0.5

1.0

1.5

a m

PS

16x83

24x123

32x163

64x243

β = 2.25



0.00 0.02 0.04 0.06 0.08 0.10 0.12a m

0.00

0.10

0.20

0.30

0.40

0.50

0.60a

mps

16x83

24x123

32x163

64x243

β = 2.25


Comparison with other studies

-0.3 0 0.3 0.6 0.9 1.2m

Q a

0

0.5

1

1.5

2

2.5

3

mπ a

βL=1.3

βL=1.7

βL=1.9

βL=2.2

βL=2.5

Pseudoscalar mass SU(2)-adjoint

[rummukainen et al 08]



0.00 0.10 0.20 0.30 0.40a m

0.00

1.00

2.00

3.00

4.00

5.00a

mps

2 /m

16x83

24x123

32x163

64x243

β = 2.25



0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40a m

0.0

2.0

4.0

6.0

8.0

10.0

mP

S/m

83 lattice

123 lattice

163 lattice


Decay constant

0.00 0.10 0.20 0.30 0.40a m

0.00

0.10

0.20

0.30a

FP

S

16x83

24x123

32x163

64x243

β = 2.25


Vector mass (1)

0.00 0.10 0.20 0.30 0.40a m

1.00

1.04

1.08

1.12

1.16m

v/mps

16x83

24x123

32x163

64x243

β = 2.25


Vector mass (2)

0.0 0.1 0.2 0.3 0.4a m

0

1

2

3

4

5

6

mV/F

PS

83 lattice

123 lattice

163 lattice


GMOR relation

m2πf

2π = Bm

0.0 0.1 0.2 0.3 0.4a m

0.0

0.1

0.2

0.3

0.4

0.5

0.6

a3 mP

S2 F

PS

2 /m

83 lattice

123 lattice

163 lattice


GMOR relation

m2πf

2π = Bm

0.00 0.02 0.04 0.06 0.08 0.10 0.12a m

0.00

0.02

0.04

0.06

0.08

a3 (Fps

mps

)2 /m

16x83

24x123

32x163

64x243

β = 2.25


GMOR relation

m2πf

2π = Bm

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07(am)

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

a3 (m

π2 Fπ2 )/

m


A hierarchy of scales

0.0 0.2 0.4 0.6 0.8 1.0a mPCAC

0.5

1.0

1.5

2.0

2.5

3.0

a M

valu

es a

t mP

CA

C =

∞

PS meson mass0

++ glueball mass

2++

glueball mass

σ1/2 (σ = string tension)


Two peculiar features: 1

1.0 1.5 2.0 2.5a mPS

0.5

1.0

1.5

2.0

2.5

3.0

a M

dynamical 0++

glueball

dynamical 2++

glueballdynamical V meson

quenched 0++

glueball

quenched 2++

glueballquenched V meson


Two peculiar features: 2

0.0 0.5 1.0 1.5 2.0 2.5 3.0a mPS

6.0

7.0

8.0

9.0

10.0

mP

S /√

σ


Conformal spectrum

QCD-like conformal

mPS

Λhad

mPS

ΛYM

[e.g. miranski 95]


Conclusions

• simulations with dynamical quarks in generic representation

• even/odd preconditioning, RHMC: arbitrary N and nf

• importance of reaching the small mass regime!!

• control over the algorithmic performance

• first preliminary results −→ larger volumes, smaller masses, scaling to controlsystematics

• improved spectroscopy −→ smearing, scalar

• NP walking behaviour: Schrödinger functional[appelquist et al 07, degrand et al 08, rummukainen et al 09]

Wilson loop, other schemes [bilgici et al 08]


Chiral condensate

chiral condensate computed from the ev distribution [giusti & luscher 08]


Running coupling

[appelquist et al 07, degrand et al 08, rummukainen et al 09]


Running mass


All-order beta function?


Systematic errors

0 5 10 15 20 25 30t

0.00

0.50

1.00

1.50

mV

,eff

24x123 lattice

32x163 lattice

64x163 lattice


Systematic errors

4 5 6 7 8t

0.00

0.20

0.40

0.60

0.80

1.00

1.20m

PS

,eff

m=-1.075m=-1.125m=-1.175

data from 83 lattice – preliminary study [kerrane 09]


Documents

Luigi Del Debbio - Swansea Universitypyweb.swan.ac.uk/~pybl/LargeNTalks/Deldebbio.pdfFlavor physics • ETC: mass term for the SM fermions, FCNC −→ ∆L ∝ 1 M2 ETC hΨΨ¯ iETC