objective: minimal realization of light composite Higgs USQCD 2015 - - PowerPoint PPT Presentation
objective: minimal realization of light composite Higgs USQCD 2015 Lattice Higgs Collaboration (L at HC) Zoltan Fodor, Kieran Holland, JK, Santanu Mondal, Daniel Nogradi, Chik Him Wong Julius Kuti University of California, San Diego USQCD
1
Zoltan Fodor, Kieran Holland, JK, Santanu Mondal, Daniel Nogradi, Chik Him Wong
R
µ
M
SCGT Theory Space close to scale invariance? nf=2 sextet rep massless fermions SU(2) doublet 3 Goldstones morph into weak bosons minimal realization
sextet rep near-conformal?
QCD intuition for near-conformal compositeness is just plain wrong Technicolor thought to be scaled up QCD motivation of the project: composite Higgs-like scalar close to the conformal window
u(+e/2 d(-e/2) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥
minimal EW embedding QCD far from scale invariance
5 10 15 20 25 −1 1 2 3 4 5 6 7 8 9 x 10
−5
t
Csinglet(t) ~ exp(-M0++·t) fitting function: Nf=12
Nf=12
Lowest 0++ scalar state from singlet correlator aM0++=0.304(18) 243x48 lattice simulation 200 gauge configs β=2.2 am=0.025
+
6 8 10 12 14 16 18 20 22 24 26 −0.5 0.5 1 1.5 2 2.5 3 x 10
−7
t
Cnon-singlet(t): Nf=12 Lowest non-singlet scalar from connected correlator aMnon-singlet = 0.420(2) !=2.2 am=0.025
−
−
−
−
n
LatHC and LatKMI Nf=12 fundamental rep
0.005 0.01 0.015 0.1 0.2 0.3 0.4 0.5 0.6
fermion mass m
triplet and singlet masses
0++ triplet state (connected) 0++ singlet state (disconnected)
From the composite Higgs mechanism: Goldstone decay constant F is setting the EWSB scale MH/F ~ 1−3 range
Triplet and singlet masses from 0
++ correlators
β=3.20 283x56, 323x64 m fit range 0.003 - 0.008
circa Lattice 2013
1 TeV a1 rho
light scalar at few hundred GeV?
EW self-energy shift
within reach of LHC Run 2 ?
t W Z
then δM2
H ⌅ 12κ2r2 t m2 t ⌅
κ2r2
t (600 GeV)2.
a0 B
near-conformal resonance spectrum separated from light scalar moving up to 2-3 TeV with refined scale setting
3 TeV
0.005 0.01 0.015 0.1 0.2 0.3 0.4 0.5 0.6
fermion mass m
triplet and singlet masses
0++ triplet state (connected) 0++ singlet state (disconnected)
From the composite Higgs mechanism: Goldstone decay constant F is setting the EWSB scale MH/F ~ 1−3 range
Triplet and singlet masses from 0
++ correlators
β=3.20 283x56, 323x64 m fit range 0.003 - 0.008
circa Lattice 2013
1 TeV a1 rho
light scalar at few hundred GeV?
EW self-energy shift
within reach of LHC Run 2 ?
t W Z
then δM2
H ⌅ 12κ2r2 t m2 t ⌅
κ2r2
t (600 GeV)2.
a0 B
near-conformal resonance spectrum separated from light scalar moving up to 2-3 TeV with refined scale setting
3 TeV
2 4 6 8 10 12 14 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25
=3.20 (with PCA analysis) 483x96 m=0.002 t effective mass of D(t) − D(T/2) MHiggs = 0.0548 ± 0.0175 2 = 0.019 Q = 1
number of blocks = 16 block size = 4 eigenvalue threshold of PCA = 0.004 error threshold of PCA = 2 fitted range = 5 − 10
2 4 6 8 10 12 14 16 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 0.3
=3.20 (with PCA analysis) 403x80 m=0.002 t effective mass of D(t) − D(T/2) MHiggs = 0.0494 ! 0.0147 2 = 0.11 Q = 0.99
number of blocks = 17 block size = 5 eigenvalue threshold of PCA = 0.004 error threshold of PCA = 2 fitted range = 4 − 10
− −
− −
−
− −
−
− −
−
0.005 0.01 0.015 0.1 0.2 0.3 0.4 0.5 0.6
fermion mass m
triplet and singlet masses
0++ triplet state (connected) 0++ singlet state (disconnected)
From the composite Higgs mechanism: Goldstone decay constant F is setting the EWSB scale MH/F ~ 1−3 range
Triplet and singlet masses from 0
++ correlators
β=3.20 283x56, 323x64 m fit range 0.003 - 0.008
circa Lattice 2013
1 TeV a1 rho
light scalar at few hundred GeV?
EW self-energy shift
within reach of LHC Run 2 ?
t W Z
then δM2
H ⌅ 12κ2r2 t m2 t ⌅
κ2r2
t (600 GeV)2.
a0 B
near-conformal resonance spectrum separated from light scalar moving up to 2-3 TeV with refined scale setting
3 TeV
2 4 6 8 10 12 14 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25
=3.20 (with PCA analysis) 483x96 m=0.002 t effective mass of D(t) − D(T/2) MHiggs = 0.0548 ± 0.0175 2 = 0.019 Q = 1
number of blocks = 16 block size = 4 eigenvalue threshold of PCA = 0.004 error threshold of PCA = 2 fitted range = 5 − 10
2 4 6 8 10 12 14 16 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 0.3
=3.20 (with PCA analysis) 403x80 m=0.002 t effective mass of D(t) − D(T/2) MHiggs = 0.0494 ! 0.0147 2 = 0.11 Q = 0.99
number of blocks = 17 block size = 5 eigenvalue threshold of PCA = 0.004 error threshold of PCA = 2 fitted range = 4 − 10
− −
− −
−
− −
−
− −
−
−
0.005 0.01 0.015 0.1 0.2 0.3 0.4 0.5 0.6
fermion mass m
triplet and singlet masses
0++ triplet state (connected) 0++ singlet state (disconnected)
From the composite Higgs mechanism: Goldstone decay constant F is setting the EWSB scale MH/F ~ 1−3 range
Triplet and singlet masses from 0
++ correlators
β=3.20 283x56, 323x64 m fit range 0.003 - 0.008
circa Lattice 2013
1 TeV a1 rho
light scalar at few hundred GeV?
EW self-energy shift
within reach of LHC Run 2 ?
t W Z
then δM2
H ⌅ 12κ2r2 t m2 t ⌅
κ2r2
t (600 GeV)2.
a0 B
near-conformal resonance spectrum separated from light scalar moving up to 2-3 TeV with refined scale setting
3 TeV
2 4 6 8 10 12 14 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25
=3.20 (with PCA analysis) 483x96 m=0.002 t effective mass of D(t) − D(T/2) MHiggs = 0.0548 ± 0.0175 2 = 0.019 Q = 1
number of blocks = 16 block size = 4 eigenvalue threshold of PCA = 0.004 error threshold of PCA = 2 fitted range = 5 − 10
2 4 6 8 10 12 14 16 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 0.3
=3.20 (with PCA analysis) 403x80 m=0.002 t effective mass of D(t) − D(T/2) MHiggs = 0.0494 ! 0.0147 2 = 0.11 Q = 0.99
number of blocks = 17 block size = 5 eigenvalue threshold of PCA = 0.004 error threshold of PCA = 2 fitted range = 4 − 10
− −
− −
−
− −
−
− −
−
−
running large volumes m fit range 0.001 - 0.002
3.2 β 2 4 6 8 10 12 14 16 18 20 22 M / F 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 M / TeV MN Ma1 Mρ
0++ scalar Higgs?
a0 scalar isovector?
strong gauge dynamics
coupling to SM?
3.2 β 2 4 6 8 10 12 14 16 18 20 22 M / F 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 M / TeV MN Ma1 Mρ
0++ scalar Higgs?
a0 scalar isovector?
strong gauge dynamics
partial compositeness ETC
?
coupling to SM?
3.2 β 2 4 6 8 10 12 14 16 18 20 22 M / F 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 M / TeV MN Ma1 Mρ
0++ scalar Higgs?
a0 scalar isovector?
strong gauge dynamics
partial compositeness ETC
?
coupling to SM?
EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)
Submitted to: Eur. Phys. J. C. CERN-PH-EP-2015-052 30th March 2015
Search for a new resonance decaying to a W or Z boson and a Higgs boson in the ``/`⌫/⌫⌫ + b¯ b final states with the ATLAS Detector
The ATLAS Collaboration Abstract
A search for a new resonance decaying to a W or Z boson and a Higgs boson in the ``/`⌫/⌫⌫+ b¯ b final states is performed using 20.3 fb1 of pp collision data recorded at ps = 8 TeV with the ATLAS detector at the Large Hadron Collider. The search is conducted by examining the WH/ZH invariant mass distribution for a localized excess. No significant deviation from the Standard Model background prediction is observed. The results are interpreted in terms of constraints on the Minimal Walking Technicolor model and on a simplified approach based
3.2 β 2 4 6 8 10 12 14 16 18 20 22 M / F 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 M / TeV MN Ma1 Mρ
0++ scalar Higgs?
a0 scalar isovector?
[GeV]
R1
m 400 600 800 1000 1200 1400 1600 1800 bb) [pb] → (H × ZH) →
1
BR (R × σ
10
10
10 1 10
ATLAS
L dt = 20.3 fb
∫
= 8 TeV s
=2 g ~ ZH →
2
,R
1
MWT R =1
v
’ HVT Benchmark model A g V Observed 95% Upper Limit Expected 95% Upper Limit 1 Sigma Uncertainty ± 2 Sigma Uncertainty ±
(a) R0
1(V00) ! ZH, H ! b¯
b
[GeV]
±
R1
m 400 600 800 1000 1200 1400 1600 1800 bb) [pb] → (H × WH) →
± 1
BR (R × σ
10
10
10 1 10
ATLAS
L dt = 20.3 fb
∫
= 8 TeV s
=2 g ~ WH →
± 2
,R
± 1
MWT R =1
v
HVT Benchmark model A g
±
V’ Observed 95% Upper Limit Expected 95% Upper Limit 1 Sigma Uncertainty ± 2 Sigma Uncertainty ±
(b) R±
1(V0±) ! WH, H ! b¯
b
[TeV]
A
m
0.5 1 1.5 2 2.5
g ~
1 2 3 4 5 6 7 8 9 10
95% CL Observed Exclusion 95% CL Expected limit Dilepton resonances 95% Exclusion Theory Inconsistent Running regime EW precision test
ATLAS
= 8 TeV s W/Z+H →
2
, R
1
R
L dt = 20.3 fb
0.002 0.004 0.006 0.008 0.01 0.012 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
m M2
linear fit range: m= 0.003 − 0.007 input from 323× 64 and 483× 96 volumes M2 = csc m +sc linear fit sc = 0.0047 ± 0.0009 csc = 8.45 ± 0.17 2/dof= 0.56
=3.20 non−Goldstone scPion spectrum
scPion ijPion Goldstone fitted not fitted
0.002 0.004 0.006 0.008 0.01 0.012 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
m M2
linear fit range: m= 0.002 − 0.006 input from 323× 64 and 483× 96 volumes M2 = csc m +sc linear fit sc = 0.0016 ± 0.0009 csc = 6.68 ± 0.22 2/dof= 1.6
=3.25 non−Goldstone scPion spectrum
scPion ijPion Goldstone fitted not fitted
chiSB in Goldstone spectrum vanishes only at zero lattice spacing decreasing lattice spacing
0.005 0.01 0.015 0.02
m at
0.01 0.02 0.03 0.04 0.05 0.06 0.07
M
2 at 2 Mij Mi5 Msc M=3.3 L^3xT = 32^3x64
M 2 ij=c0 + c1 m c0=-3.753e-04 +/- 5.33e-03 c1=6.310e+00 +/- 7.25e-01new runs here
0.002 0.004 0.006 0.008 0.01 0.012 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
m M2
linear fit range: m= 0.003 − 0.007 input from 323× 64 and 483× 96 volumes M2 = csc m +sc linear fit sc = 0.0047 ± 0.0009 csc = 8.45 ± 0.17 2/dof= 0.56
=3.20 non−Goldstone scPion spectrum
scPion ijPion Goldstone fitted not fitted
idea:
generated with sea fermions
unnecessarily complicated
gradient flow
valence spectrum
with original standard analysis when cutoff is removed: cross check
0.002 0.004 0.006 0.008 0.01 0.012 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
m M2
linear fit range: m= 0.002 − 0.006 input from 323× 64 and 483× 96 volumes M2 = csc m +sc linear fit sc = 0.0016 ± 0.0009 csc = 6.68 ± 0.22 2/dof= 1.6
=3.25 non−Goldstone scPion spectrum
scPion ijPion Goldstone fitted not fitted
chiSB in Goldstone spectrum vanishes only at zero lattice spacing decreasing lattice spacing
0.002 0.004 0.006 0.008 0.01 0.012 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
m M2
linear fit range: m= 0.003 − 0.007 input from 323× 64 and 483× 96 volumes M2 = csc m +sc linear fit sc = 0.0047 ± 0.0009 csc = 8.45 ± 0.17 2/dof= 0.56
=3.20 non−Goldstone scPion spectrum
scPion ijPion Goldstone fitted not fitted
idea:
generated with sea fermions
unnecessarily complicated
gradient flow
valence spectrum
with original standard analysis when cutoff is removed: cross check
0.002 0.004 0.006 0.008 0.01 0.012 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
m M2
linear fit range: m= 0.002 − 0.006 input from 323× 64 and 483× 96 volumes M2 = csc m +sc linear fit sc = 0.0016 ± 0.0009 csc = 6.68 ± 0.22 2/dof= 1.6
=3.25 non−Goldstone scPion spectrum
scPion ijPion Goldstone fitted not fitted
chiSB in Goldstone spectrum vanishes only at zero lattice spacing decreasing lattice spacing
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
m M2
inputs from volume 403! 80 flow time t=2 linear fit =3.2 M2
= 2B m
2B = 1.243 " 0.0041 2/dof = 0.126
sextet model Goldstone pion corner pion channel flow time t=2
cyan: scPion overlay
fitted linear fit
chiral symmetry restored
20 40 60 80 100 120 0.05 0.1
=3.20 323× 64 m=0.004 gradient flow time t=3 120 eigenvalues
Quartets of Q=−1 Dirac spectrum on gradient flow eigenvalue multiplets staggered quartet eigenvalue spectrum
degenerate quartets⇒valence chiral symmetry restored
0.005 0.01 0.015 0.02 0.025 0.03 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
valence mass m F
m fit range: 0.008 − 0.024 483× 96 NLO linear fit =3.20 m = 0.003 original ensemble F = F + pF m F = 0.0204 ± 7.5e−05 pF = 0.351 ± 0.0043 2/dof = 0.0197
sextet mixed action at flow time t=2 F (rwall pion channel)
m fit range: 0.008 − 0.024 483× 96 NLO linear fit =3.20 m = 0.003 original ensemble F = F + pF m F = 0.0204 ± 7.5e−05 pF = 0.351 ± 0.0043 2/dof = 0.0197
fitted linear fit
new range
0.5 1 1.5 2 1 2 3 4 5 <q-
v qv> / !
MvvL Nf = 2, mu=10 MeV, L=2 fm new formula ("=0) p-expansion (#=0) $-expansion ("=0) 0.5 1 1.5 2 5 10 15 20 25 30 %&"(')/ ! '!V Nf = 2, L=2 fm, mu=10 MeV new formula ("=0) p-expansion (#=0) $-expansion ("=0)
Damgaard and Fukaya Damgaard and Fukaya
new analysis in crossover and RMT regime opens up with mixed action on gradient flow
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10
−3eigenvalue scale
spectral density (,m)
Vol=403! 80 =3.25 m=0.002 25 configurations with Q=0 300 eigenvalues
flow time t=3
spectral density (,m)
1 2 3 4 5 6 7 x 10
−30.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10
−3eigenvalue scale
spectral density (,m)
Vol=483× 96 =3.25 m=0.002 37 configurations with Q=0 300 eigenvalues flow time t=3 spectral density (,m)
RMT regime
RMT regime
0.5 1 1.5 2 1 2 3 4 5 <q-
v qv> / !
MvvL Nf = 2, mu=10 MeV, L=2 fm new formula ("=0) p-expansion (#=0) $-expansion ("=0) 0.5 1 1.5 2 5 10 15 20 25 30 %&"(')/ ! '!V Nf = 2, L=2 fm, mu=10 MeV new formula ("=0) p-expansion (#=0) $-expansion ("=0)
Damgaard and Fukaya Damgaard and Fukaya
new analysis in crossover and RMT regime opens up with mixed action on gradient flow
Σ is not RG invariant, requires rescaling
to reach RMT regime close to CW requires large resources
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10
−3eigenvalue scale
spectral density (,m)
Vol=403! 80 =3.25 m=0.002 25 configurations with Q=0 300 eigenvalues
flow time t=3
spectral density (,m)
1 2 3 4 5 6 7 x 10
−30.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10
−3eigenvalue scale
spectral density (,m)
Vol=483× 96 =3.25 m=0.002 37 configurations with Q=0 300 eigenvalues flow time t=3 spectral density (,m)
RMT regime
RMT regime
0.5 1 1.5 2 1 2 3 4 5 <q-
v qv> / !
MvvL Nf = 2, mu=10 MeV, L=2 fm new formula ("=0) p-expansion (#=0) $-expansion ("=0) 0.5 1 1.5 2 5 10 15 20 25 30 %&"(')/ ! '!V Nf = 2, L=2 fm, mu=10 MeV new formula ("=0) p-expansion (#=0) $-expansion ("=0)
Damgaard and Fukaya Damgaard and Fukaya
new analysis in crossover and RMT regime opens up with mixed action on gradient flow
0.5 1 1.5 2 1 2 3 4 5 6 7
eigenvalue scale
2() spectral density 483× 96 beta=3.20 m=0.002 LatHC
63,700,992 eigenvalues in D+D (Chebyshev expansion) = 2(0) Banks−Casher relation (nf=2)
topology: Q=0 spectral density of full Dirac spectrum (sextet rep)
0.5 1 1.5 2 1 2 3 4 5 6 7 x 10
7
scale of mode number count
(,m ) 483× 96 beta=3.20 m=0.002 LatHC
63,700,992 eigenvalues in D+D (,m ) = 2V
0 (,m ) mode number distribution
403× 80 data scaled with volume: blue dots
exact sum
mode number of full Dirac spectrum (sextet rep)
Improved determination of the chiral condensate Σ compared from Dirac spectra and the Chebyshev expansion. With the additive NLO cutoff term separated from B and new fit to F, the improved result on Σ eliminates previous discrepancies in the GMOR relation.
GMOR relation (nf=2): 2BF2 = Σ (Σ is the chiral condensate) F: decay constant of Goldstone pion Mπ
2 = 2B⋅m in LO χPT
Σeff Σ = 1+ Σ 32π3NFF4 2N2
F|Λ|arctan |Λ|
m 4π|Λ|N2
Fmlog Λ2 +m2
µ2 4mlog |Λ| µ
0.5 1 1.5 2 2.5 3 x 10
−30.005 0.01 0.015
eigenvalue scale
spectral density (,m)
Vol=483! 96 =3.20 m=0.002 9 configurations with Q=0 171 eigenvalues Chebyshev expansion: cyan(1K) to green(4K) direct eigenvalue spectrum (red) spectral density (,m)
0.5 1 1.5 2 2.5 3 x 10
−30.005 0.01 0.015
eigenvalue scale
spectral density (,m)
Vol=483× 96 =3.20 m=0.002 9 configurations with Q=−1 180 eigenvalues Chebyshev expansion 3K:cyan, 6K:magenta direct eigenvalue spectrum (blue) spectral density (,m)
0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
scale of anomalous mass dimension
(,m) anomalous mass dimension 483× 96 (magenta) =3.20 m=0.003 LatHC −1 ~ footprint of gradient flow?
4−loop (RS)
483× 96 =3.20 m=0.002 data: blue circles
anomalous mass dimension from full Dirac spectrum (sextet rep)
0.5 1 1.5 2 1 2 3 4 5 6 7 x 10
7
scale of mode number count
(,m ) 483× 96 beta=3.20 m=0.002 LatHC
63,700,992 eigenvalues in D+D (,m ) = 2V
0 (,m ) mode number distribution
403× 80 data scaled with volume: blue dots
exact sum
mode number of full Dirac spectrum (sextet rep)
Boulder group pioneered fitting procedure
ν R(M R,mR) = ν(M,m) ≈ const ⋅ M
4 1+γ m (M ),
4 1+γ m (λ) , with γ m(λ) fitted
0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
scale of anomalous mass dimension
(,m) anomalous mass dimension 483× 96 (magenta) =3.20 m=0.003 LatHC −1 ~ footprint of gradient flow?
4−loop (RS)
483× 96 =3.20 m=0.002 data: blue circles
anomalous mass dimension from full Dirac spectrum (sextet rep)
0.5 1 1.5 2 1 2 3 4 5 6 7 x 10
7
scale of mode number count
(,m ) 483× 96 beta=3.20 m=0.002 LatHC
63,700,992 eigenvalues in D+D (,m ) = 2V
0 (,m ) mode number distribution
403× 80 data scaled with volume: blue dots
exact sum
mode number of full Dirac spectrum (sextet rep)
Boulder group pioneered fitting procedure
ν R(M R,mR) = ν(M,m) ≈ const ⋅ M
4 1+γ m (M ),
4 1+γ m (λ) , with γ m(λ) fitted
1 2 3 4 5 6 7 8 9 10 11 12
g
2
0.2 0.4 0.6 0.8
m
6 = 0 6 = 0.5
DeGrand, Shamir, Svetitsky
1 2 3 4 5 6 7 8 9 10 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
gradient flow time t g2(t,m2)
Wilson flow, Symanzik action 6/g0
2 = 3.25
483x96 m = 0.002−0.004 323x64 m = 0.005−0.006,0.008
scale−dependent running coupling on gradient flow
6.6 6.8 7 7.2 7.4 7.6 7.8 8 2 4 6 8 10 12 14 16
renormalized gauge coupling g2 t0 scale of g2 in chiral limit
6/g2
0=3.25
6/g2
0=3.20
cubic spline interpolation
renormalized gauge coupling in chiral limit
running coupling at fixed β
2 is linear
scale change at fixed renormalized g2 a → 0 needed
nstructed at flow time 0 depend on t
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 1 2 3 4 5 6 7 8 9 10
M2
6/g0
2 = 3.25
m=0.003−0.006,0.008 fitted Chiral PT fits excellent 2 fits
t0 scale of selected g2 series in m=0 chiral limit
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 2 3 4 5 6 ( g2(sL) - g2(L) ) / log( s2 ) g2(L) Nf = 2 sextet continuum c=7/20 SSC 1 loop 2 loop
LatHC Nf = 2 sextet
monotonic increase of g2 with scale is consistent with:
lattice step functions: 12➞18, 16➞24, 20➞30, 24➞36 last two step functions are critical in the analysis: SSC vs. WSC are consistent at large flow times which requires the large volumes
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 2 3 4 5 6 ( g2(sL) - g2(L) ) / log( s2 ) g2(L) Nf = 2 sextet continuum c=7/20 SSC 1 loop 2 loop
LatHC Nf = 2 sextet
Nf = 2 sextet
0.1 0.2 0.3 0.4 0.5
u = 1/g
2
0.05
6 = 0 6 = 0.5 linear extrap quad extrap
two loops
DeGrand, Shamir, Svetitsky
monotonic increase of g2 with scale is consistent with:
lattice step functions: 12➞18, 16➞24, 20➞30, 24➞36 last two step functions are critical in the analysis: SSC vs. WSC are consistent at large flow times which requires the large volumes
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 2 3 4 5 6 ( g2(sL) - g2(L) ) / log( s2 ) g2(L) Nf = 2 sextet continuum c=7/20 SSC 1 loop 2 loop
LatHC Nf = 2 sextet
Nf = 2 sextet
0.1 0.2 0.3 0.4 0.5
u = 1/g
2
0.05
6 = 0 6 = 0.5 linear extrap quad extrap
two loops
DeGrand, Shamir, Svetitsky
monotonic increase of g2 with scale is consistent with:
lattice step functions: 12➞18, 16➞24, 20➞30, 24➞36 last two step functions are critical in the analysis: SSC vs. WSC are consistent at large flow times which requires the large volumes
0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9
β(g2) g2
2-loop perturb. 4-loop MS continuum 104 124 144 164
Boulder-Tel Aviv Nf = 2 sextet without step functions 20➞30, 24➞36 Wilson flow only
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 2 3 4 5 6 ( g2(sL) - g2(L) ) / log( s2 ) g2(L) Nf = 2 sextet continuum c=7/20 SSC 1 loop 2 loop
LatHC Nf = 2 sextet
Nf = 2 sextet
0.1 0.2 0.3 0.4 0.5
u = 1/g
2
0.05
6 = 0 6 = 0.5 linear extrap quad extrap
two loops
DeGrand, Shamir, Svetitsky
monotonic increase of g2 with scale is consistent with:
lattice step functions: 12➞18, 16➞24, 20➞30, 24➞36 last two step functions are critical in the analysis: SSC vs. WSC are consistent at large flow times which requires the large volumes
0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9
β(g2) g2
2-loop perturb. 4-loop MS continuum 104 124 144 164
Boulder-Tel Aviv Nf = 2 sextet without step functions 20➞30, 24➞36 Wilson flow only
This is disagreement is between two analyses nothing to do with staggered formulation
besides: promoting a beta function zero to conformal IRFP would require to remove the cutoff with the ω scaling exponent
17
Three SU(3) sextet fermions can give rise to a color singlet. The tensor product 6⌦6⌦6 can be decomposed into irreducible representations of SU(3) as, 6⌦6⌦6 = 12⇥810103⇥27282⇥35 where irreps are denoted by their dimensions and 10 is the complex conjugate of 10. Fermions in the 6-representation carry 2 indices, ψab, and transform as ψaa0 ! Uab Ua0b0 ψbb0 and the singlet can be constructed explicitly as εabc εa0b0c0 ψaa0 ψbb0 ψcc0.