Luigi Del Debbio - School of Physics and Astronomyldeldebb/zeuthen10.pdfLattice studies of the conformal window Luigi Del Debbio University of Edinburgh Zeuthen - November 2010 T H

Lattice studies of the conformal window

Luigi Del Debbio

University of Edinburgh

Zeuthen - November 2010T

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UN I V E

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Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November 2010 1 / 34

Outline

1 Introduction

2 RG flow

3 Scaling laws

4 Schrodinger functional

5 Outlook


IntroductionTechnicolor models

ΛTC

METC

ΛQCD

ΛTC

<ΨΨ> =3

TC

ETC

SM

flavor

gauge GeV

TeV

???


IntroductionFlavor constraints for TC

ETC: mass term for the SM fermions, FCNC

−→ ∆L ∝ 1

M2ETC

〈ΨΨ〉ETC ψψ,1

M2ETC

ψψ ψψ

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

〈ΨΨ〉ETC = 〈ΨΨ〉TC exp

(∫ METC

ΛTC

dµ

µγ(µ)

)[holdom, lane, appelquist, luty, sannino, chivukula,...]


IntroductionInfrared fixed points

[Banks & Zaks 82]

β(g) = µd

dµg(µ) = −β0g

3 − β1g5 + O(g 7)

β0 > 0 =⇒ asymptotic freedom, nf < naff

β(g) < 0 =⇒ confining theories

β(g∗) = 0 =⇒ IR fixed point

conformal window: IR zero of the β function

ncf < nf < naf

f

(4π)2β0 (4π)4β1

SU(3), nf = 3 fund 9 64SU(3), nf = 12 fund 3 -50SU(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-loop

2-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-loop

2-loop


IntroductionThe scales of an IRFP

g

ΛU

ΛIR

g*

µ

asymptotic freedom at high-energy scale ΛUV

scale invariance at large distances =⇒ no massive spectrum

large-distance dynamics: anomalous dimensions at the IRFP

nonperturbative IRFP: strong coupling, large anomalous dimensions

scale invariance explicitly broken by a mass term/finite volume


IntroductionThe conformal window

Fund

2A

2S Adj

Ladder

γ = 1 γ = 2

Ryttov & Sannino 07

SU(N) Phase Diagram

Dietrich & Sannino 07

Sannino & Tuominen 04


RG flowRG transformation

Action with a cutoff µ: S(gi ) =∑

i giµ4−diOi

RG transformation: µ 7→ µ′ = µ/b – at constant physics

g ′i = Ri (g)

↪→ flow in the (infinite-dimensional) parameter space.

Fixed point:

g∗i = Ri (g∗)

In the vicinity of the fixed point:

δg ′i =∂Ri

∂gk

∣∣∣∣∗δgk

v ′i = byi vi

yi characterize the fixed pointyi > 0 relevant parameters


RG flowRG flow for QCD-like theories

YM theory - g lattice bare coupling, g ′ space of irrelevant couplings

g

g’


RG flowRG flow for conformal theories

flow in the m = 0 plane - assuming only the mass is relevant in the IR

g

g’


RG flowRG flow for conformal theories

flow in the presence of a mass deformation

(g,g’)

am

?


Scaling lawsMeson spectrum

[luty 09, degrand09, ldd&zwicky 10]

CH(t; g ,m, µ) =

∫d3x 〈H(x , t)H(0)†〉

∼ F 2H e−MH t

RG transformation:

ZH(µ)2CH(t; g ,m, µ) = ZH(µ′)2CH(t; g ′,m′, µ′)

CH(t; g ,m, µ) = b−2γHCH(t; byg g , bymm, µ′)

naive dimensional analysis:

CH(t; byg g , bymm, µ′) = b−2dHCH(tb−1; byg g , bymm, µ)

choosing b such that bymm = 1:

CH(t; g ,m, µ) = m2∆H/ymF(tm1/ym ) , ∆H = dH + γH

MH ∼ m1/(1+γ∗)


Scaling lawsScaling of matrix elements

[ldd&zwicky 10]

Tφ1Oφ2 (g ,m, µ) = 〈φ1|O|φ2〉|g,m,µRG transformation:

Tφ1Oφ2 (g ,m, µ) = b−γOTφ1Oφ2 (g ′,m′, µ′)

= b−γOTφ1Oφ2 (byg g , bymm, µ/b)

naive dimensional analysis:

Tφ1Oφ2 (byg g , bymm, µ/b) = b−dO+d1+d2Tφ1Oφ2 (bymm, µ)

choosing b such that bymm = 1

Tφ1Oφ2 (g ,m, µ) ∼ m(∆O+d1+d2)/ym


Scaling lawsScaling from the trace anomaly

trace anomaly

θαα =β

2G 2 + nf (1 + γm)mqq

matrix elements of the energy-momentum tensor

〈H(p)|θαβ |H(p)〉 = 2pαpβ

hence we can write the trace at the fixed point

2M2H = nf (1 + γm)〈H(p)|mqq|H(p)〉

using the scaling law for the matrix element

MH ∼ m1/ym


Scaling lawsConformal spectrum

In the presence of a mass deformation (and infinite volume)

no parametric separation between the PS and the other channels

all masses/decay constants scale like m1/ym (but M ∼ F (0)/L !!)

↪→ very different spectrum from a QCD-like theory [Miransky 94]

QCD-like conformal

mPS

Λhad

mPS

ΛYM


SU(2) adjointExploring the chiral regime of MWT

[catterall et al 07, ldd et al 08/10, rummukainen et al 08]

MC simulations with nf = 2 Wilson fermions

-1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8m

0

1.8

2.0

2.2

2.4

2.6

2.8

3.0

β

Catterall et al.

Hietanen et al.

Del Debbio et al.


SU(2) adjointChiral limit

-1.20 -1.15 -1.10a m

0

0.00

0.10

0.20a m

16x83

24x123

64x243

32x163

β = 2.25

a mc = -1.202(1)


SU(2) adjointPseudoscalar mass - 1

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

0.60

a m

ps

16x83

24x123

32x163

64x243

β = 2.25



0.00 0.10 0.20 0.30 0.40a m

1.00

2.00

3.00

4.00

5.00

a m

ps

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 0.40

a m

0.0

2.0

4.0

6.0

8.0

10.0

mP

S/m

83 lattice

123 lattice

163 lattice

243 lattice

fit

logmPS =1

1 + γlogm + C = 0.85 logm + C


SU(2) adjointVector mass

0.00 0.10 0.20 0.30 0.40a m

0.96

1.00

1.04

1.08

1.12

1.16

1.20m

v/m

ps

16x83

24x123

32x163

64x243

β = 2.25


SU(2) adjointGluonic sector - 1

0 2 4 6 8 10 12 14R

0.00

0.05

0.10

0.15

0.20

0.25

V(R

)

24x123

32x163

fit with fixed σ

fit with 1/R3 term

fit with variable σ

am=0.12, amPS

=0.6



0.0 0.2 0.4 0.6 0.8 1.0a m

PCAC

0.5

1.0

1.5

2.0

2.5

3.0a

M

va

lue

s a

t m

PC

AC =

∞

PS meson mass

0++

glueball mass

2++

glueball mass

σ1/2

(σ = string tension)



0.0 0.5 1.0 1.5 2.0 2.5 3.0

a mPS

4.0

5.0

6.0

7.0

8.0

9.0

10.0

mP

S /√

σ


Scaling lawsFinite-size scaling

RG transformationCH(t; m, L, µ) = b−2γHCH(t; m′, L, µ′) ,

naive dimensional analysis

CH(t; m, L, µ) = b−2(dH+γH )CH(b−1t; bymm, b−1L, µ) .

choosing b such that b−1L = L0

CH(t; m, L, µ) =

(L

L0

)−2∆H

CH

(t

L/L0; x

1

µLym0

, L0, µ

),

where the scaling variable is x = Lymm.

MH = L−1f (x) ,

f (x)→ 0 when x → 0m scaling in the thermodynamic limit

f (x) ∼ x1/ym , as x →∞ .


SU(2) adjointFinite-size scaling

1

2

3L

FP

SL= 8

L=12

L=16

1 2 3 4 5 6

x

1

2

3

LF

PS

1 2 3 4 5 6

x

γ*=0.1

γ*=0.4 γ

*=1.0

γ*=0.2


Scaling lawsScaling of the chiral condensate

RG analysis〈qq〉 ∼ mηqq

where

ηqq =∆qq

ym=

3− γ∗1 + γ∗

relation to the eigenvalue density

〈qq〉 = −2

∫ µ/m

0

dxρ(mx)

1 + x2+A(m) ,

〈qq〉 m→0∼ mηqq ⇔ ρ(λ)λ→0∼ ληqq .

This in turn implies:ηqq|QCD−like = 0 , ηqq|mCGT > 0 ,


SU(2) adjointscaling of the eigenvalues

ν(M,m) =

∫ +Λ

−Λ

dλ ρ(λ) , Λ =√

M2 −m2

ν(Λ, 0) = C(Λ− g)ηqq+1

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180

20

40

60

80

100

mode number distribution h2stEntries 18

Mean 0.01406

RMS 0.004063

/ ndf 2χ 5.801 / 8

Prob 0.6695

p0 9.964e+03± 7.038e+04

p1 0.0001830± 0.0002312

p2 0.037± 1.652

h2stEntries 18

Mean 0.01406

RMS 0.004063

/ ndf 2χ 5.801 / 8

Prob 0.6695

p0 9.964e+03± 7.038e+04

p1 0.0001830± 0.0002312

p2 0.037± 1.652

mode number distribution

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180

20

40

60

80

100

mode number distribution h2stEntries 18

Mean 0.01406

RMS 0.004063

/ ndf 2χ 0.5054 / 2

Prob 0.7767

p0 2.411e+05± 2.373e+05

p1 0.000803± 0.001256

p2 0.250± 1.952

h2stEntries 18

Mean 0.01406

RMS 0.004063

/ ndf 2χ 0.5054 / 2

Prob 0.7767

p0 2.411e+05± 2.373e+05

p1 0.000803± 0.001256

p2 0.250± 1.952

mode number distribution


Schrodinger functionalRenormalized coupling

Schrodinger functional [ALPHA collaboration]

Z [η] = e−Γ[η] =

∫L×L3

DUDψDψ exp(−S [U, ψ, ψ])

Dirichlet boundary conditions at t = 0, L, dependent on η

k

g 2(L)=

∂Γ

∂η

∣∣∣∣η=0

, g 2 = g 20 + O(g 4

0 )

Lattice step scaling function

Σ(u, a/L) = g 2(bL)∣∣∣g2(L)=u,m=0

Step scaling functionσ(u) = lim

a→0Σ(u, a/L)

fixed point: σ(u∗) = u∗


Schrodinger functionalRenormalized mass

Definition of the renormalized mass

∂µ(AR)µ = 2mPR

(AR)µ(x) = ZAψ(x)γµγ5ψ(x)

(PR)(x) = ZP ψ(x)γ5ψ(x)

Lattice step scaling function

ZP(g0, L/a) = c

√3f1

fP(L/2)

ΣP(u, a/L) =ZP(g0, bL/a)

ZP(g0, L/a), g 2(L) = u

Step scaling functionσP(u) = lim

a→0ΣP(u, a/L)


Schrodinger functionalScheme-dependence

existence of the IRFP is scheme-independent

β(g) = µd

dµg(µ)

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

β′(g ′) =∂

∂gΦ(g)β(g)

universality of the first two coefficients

β-function away from IRFP is not universal!

m′ = mF (g)

γ′ = γ + β∂

∂glog F


Schrodinger functionalSU(2) adjoint

nf = 2 Wilson, Schrodinger functional coupling [Hietanen et al 08, Bursa et al 09]

0.0 1.0 2.0 3.0u

0.98

1.00

1.02

1.04

1.06

1.08

1.10

σ(u

)/u

1-loop

2-loop

Statistical

∆(L1, L2) = 1/g 2(L2)− 1/g 2(L1)


ResultsSU(2) adjoint

nf = 2 Wilson, Schrodinger functional mass [Bursa et al 09]

0 0.5 1 1.5 2 2.5 3 3.5u

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04σ

P(4

/3,u

)

1-loop

Statistical Error

log |σP(u, b)| = − γ log b =⇒ 0.05 < γ < 0.56


OutlookTowards phenomenology

tools have been developed and tested - keep looking for better tools

preliminary results need to be put on solid groundbenchmark codes/analysiscontrol systematics: small masses & large volumesimproved actions

all methods have systematicsmore confident when different approaches yield consistent results

anomalous dimensions are small

SU(2) adjoint evidence for conformality

deformations away from conformalitynf < ncfmass deformation4fermi interactions

lattice results to be taken into account for model building

talk to phenomenologists!


Documents

Luigi Del Debbio - School of Physics and Astronomyldeldebb/zeuthen10.pdfLattice studies of the conformal window Luigi Del Debbio University of Edinburgh Zeuthen - November 2010 T H