IR fixed points in gauge theories from lattice simulations Luigi - - PowerPoint PPT Presentation
IR fixed points in gauge theories from lattice simulations Luigi Del Debbio Higgs Centre for Theoretical Physics - University of Edinburgh Nagoya, March 2014 1 Plan RG flows, fixed points, anomalous dimensions Anomalous
Higgs Centre for Theoretical Physics - University of Edinburgh
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Nagoya, March 2014
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Dependence of the dimensionless couplings on the cut-off
in the couplings (and rescaling of the fields).
compatible with the symmetries of the system.
k
Consider a theory with a UV cutoff - integrate out UV modes:
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An RG transformation can be seen as a flow in the space of couplings.
$z-
k(ˆ
k
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Long-distance dynamics is described by
'- o|'-^v
The couplings may have a finite limit in the IR, and flow towards fixed point values: Scale invariant theory at the fixed point
µ→0 ˆ
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Linearized RG flow in a neighbourhood of a fixed point Associated IR scale: Stable hierarchy generated by weakly relevant operators. [Strassler 03, Sannino 04, Luty&Okui 04] YM theory at the GFP is a limiting case:
ΛIR ∼ ΛUV exp{− 1 β0g2 }
Global-singlet relevant operators (GSRO) require fine-tuning.
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In the SM + elementary Higgs: dimension = 1+3 = 4 In DEWSB: scalar is composite [Dimopoulos et al 79, Eichten et al 1980] dimension = 3+3 = 6 Tension with suppressing FCNC
f Λ2
UV
¯ qq¯ qq
dimension = 6
dim(H†H) ' 2
GSRO
UV
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Alleviate the problem due to the large dimension of the composite scalar Theory at the EW scale is near a non-trivial fixed point Scaling dimension of the fermion bilinear is smaller
dim(H) small, but dim(H† H) > 2 dim(H)
[Holdom,Yamawaki, Appelquist, Eichten, Lane] [Sannino 04, Luty 04, Rattazzi et al 08]
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Under axial transformations in flavor space:
Flavor non-singlet lies in the same multiplet Construct a basis of operators that is closed under renormalization:
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Color trace identity: Fierz transformation:
1 )ij(Γ(r) 2 )kl =
s
1 )il(Γ(s) 2 )kj
1 T Aψ2)( ¯
2 T Aψ4) 7!
1 ψ2)( ¯
2 ψ4) +
s
1 ψ4)( ¯
2 ψ2)
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Γ1Γ2 = 1
1 = O± V V +AA
2 = O± V V −AA
3 = O± SS−P P
4 = O± SS+P P
5 = O± T T
i,R = Z± ij
jk
k
±
scale-independent
[Donini et al 99, Guagnelli et al 05]
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Γ1Γ2 = 1
1 = O± V A+AV
2 = O± V A−AV
3 = −O± SP −P S
4 = O± SP +P S
5 = O± T ˜ T
± R
±
±
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Finite-volume renormalization scheme:
The renormalized charge is an observable, i.e. it can be measured by numerical simulations.
yields a nonperturbative coupling.
system. g2(L) = k ⌧∂S ∂η −1
[M Luscher et al]
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The running of the coupling as the scale is varied by a factor s is encoded in the step scaling function: Σ(u, s, a/L) = g2(g0, sL/a)
Lattice step scaling is affected by lattice artefacts, i.e. depends on the details of the UV
σ(u, s) = lim
a/L→0 Σ(u, s, a/L)
−2 log s = Z σ(u,s)
u
dx √xβ(√x) L/a: resolution of the simulation σ(u, s) = u ⇐ ⇒ σ(u, s)/u = 1 At the fixed point:
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The renormalized mass is defined as: ¯ m(µ) = ZA ZP (µ)m In order to study its running we need to compute nonperturbatively: ZP (L) = p 3f1/fP (L/2) f1 = 1/12L6 Z d3u d3v d3y d3z hζ
0(u)γ5τ aζ0(v)ζ(y)γ5τ aζ(z)i ,
fP (x0) = 1/12 Z d3y d3z hψ(x0)γ5τ aψ(x0)ζ(y)γ5τ aζ(z)i .
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Step scaling functions for the mass: ΣP (u, s, a/L) = ZP (g0, sL/a) ZP (g0, L/a)
σP (u, s) = lim
a→0 ΣP (u, s, a/L)
σP (u) = ✓ u σ(u) ◆(d0/(2β0)) exp "Z √
σ(u) √u
dx ✓γ(x) β(x) − d0 β0x ◆# Relation to the anomalous dimension:
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In the neighbourhood of the fixed point: Z m(µ/s)
m(µ)
dm m = −γ∗ Z µ/s
µ
dq q log |σP (s, u)| = −γ∗ log s Hence we can define an estimator for the anomalous dimension: ˆ γ(u) = −log |σP (u, s)| log |s| Scheme-independent in a neighbourhood of a fixed point.
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i;A,B,C = 1
53[ΓC] Q± i O21[ΓA] O45[ΓB]i
f1f2[Γ] = a6 X y,z
f1(y)Γζ0 f2(z)
y,z
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12[γ5] O21[γ5]i
k=1,2,3
12[γk] O21[γk]i
i;A,B,C(x0) =
i;A,B,C(x0)
1 k3/2−η 1
1 (g0, aµ) h± 1;A,B,C(L/2) = h± 1;A,B,C(L/2)
1 = h1;γ5γ5γ5
2 = 1
j,k,l
3 = 1
k
4 = 1
k
5 = 1
k
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Step-scaling functions:
g(L)2=u
a→0 Σ±(s; u, L/a) = T exp
σ(u) √u
In a neighbourhood of a fixed point:
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Use lattice tools to search for IRFPs in 4D SU(N) gauge theories
Fund 2A 2S Adj
Ladder
Catterall, Sannino Del Debbio, Patella,Pica Catterall, Giedt, Sannino, Schneible Iwasaki et al.
Appelquist, Fleming, Neil Deuzeman, Lombardo, Pallante
× ×
γ = 1 γ = 2
Shamir, Svetitsky, DeGrand
Ryttov and F.S. 07 All Orders Beta Function
[Yamawaki, Appelquist, Miransky, Schrock, Nunez, Piai, Hong, Braun, Gies]
MWT
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0.0 1.0 2.0 3.0 u 0.98 1.00 1.02 1.04 1.06 1.08 1.10 σ(u)/u 1-loop 2-loop Statistical
[Bursa et al 09]
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0.5 1 1.5 2 2.5 3 3.5 u 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 σP(4/3,u) 1-loop Statistical Error
γ(u) = −log |σP (u, s)| log |s| γ < 0.6
[Bursa et al 09]
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0.2 0.4 0.6 0.8 1 1 2 3 4 5 6
L/a=8, sL/a=16 L/a=10, sL/a=16 L/a=8, sL/a=12 1-loop
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0.2 0.4 0.6 0.8 1 1 2 3 4 5 6
1
+
2
+
3
+
4
+
5
+
1
1 loop +
Multiplicative renormalization in this channel Scheme-independence: system in a neighbourhood of a fixed point?
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Checks with different scaling steps
0.2 0.4 0.6 0.8 1 1 2 3 4 5 6
1
+
2
+
3
+
4
+
5
+
1
1 loop +
0.2 0.4 0.6 0.8 1 1 2 3 4 5 6
1
+
2
+
3
+
4
+
5
+
1
1 loop +
Results are consistent
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Goal: robust evidence for an IR fixed point in SU(2) adj Determination of the critical exponents at the IRFP using SF mass anomalous dimension & 4fermi anomalous dimensions are feasible new couplings: gradient flow, chirally rotated [Ramos 13, Sint 10] Comparison with other methods - scaling of spectral quantities
Energy-momentum tensor and IRFP