Post on 19-Mar-2021
Lattice studies of the conformal window
Luigi Del Debbio
University of Edinburgh
Nagoya - August 2011T
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Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 1 / 34
Outline
1 Introduction
2 RG flow
3 Scaling laws
4 Schrodinger functional
5 Outlook
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 2 / 34
IntroductionTechnicolor models
ΛTC
METC
ΛQCD
ΛTC
<ΨΨ> =3
TC
ETC
SM
flavor
gauge GeV
TeV
???
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 3 / 34
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,...]
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 4 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 5 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 6 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 7 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 8 / 34
RG flowRG flow for QCD-like theories
YM theory - g lattice bare coupling, g ′ space of irrelevant couplings
g
g’
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 9 / 34
RG flowRG flow for conformal theories
flow in the m = 0 plane - assuming only the mass is relevant in the IR
g
g’
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 10 / 34
RG flowRG flow for conformal theories
flow in the presence of a mass deformation
(g,g’)
am
?
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 11 / 34
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∗)
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 12 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 13 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 14 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 15 / 34
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.
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 16 / 34
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)
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 17 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 18 / 34
SU(2) adjointPseudoscalar mass - 2
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 19 / 34
SU(2) adjointPseudoscalar mass - 3
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 20 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 21 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 22 / 34
SU(2) adjointGluonic sector - 2
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)
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 23 / 34
SU(2) adjointGluonic sector - 3
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 24 / 34
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 →∞ .
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 25 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 26 / 34
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 ,
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 27 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 28 / 34
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∗
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 29 / 34
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)
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 30 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 31 / 34
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)
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 32 / 34
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
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 33 / 34
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!
Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Nagoya - August 2011 34 / 34