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Lattice simulations and extensions of thestandard model with strong interactions
Georg BergnerFSU Jena
SIFT 2017: Jena 23. 11. 2017
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1 Strong interactions
2 Supersymmetric Yang-Mills theory on the lattice
3 Adjoint QCD and Technicolour theories
4 Conclusions and outlook
in collaboration with P. Giudice, S. Kuberski, G. Munster, I. Montvay,
S. Piemonte, S. Ali, H. Gerber, P. Scior, C. Lopez
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Strong interactionsFeatures of strong interactions:
formations of bound states of hadronsand nuclear matter
create most of the mass of the bayonicmatter in the universe
dynamical scale generation
gu
uQ >> ΛQCD Q << ΛQCD
u d
uu
d
ggg
u + u + d −→ uu
d
3× 5 MeV 1000 MeV ∝ ΛQCD
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The extension/completion/explanation of the StandardModel
Why extend the standard model of particle physics:
add missing explanation for Dark Matter
solve hierarchy problem and explain small Higgsmass
explain standard model parameters: flavour,mass parameters / Yukawa couplings
may other motivations: neutrino physics,quantum gravity, unification of forces
dark energy
dark matter
SM matter
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The extension/completion/explanation of the StandardModel
General paradigms for the introduction of new physics:
1 new physics means new parameters and scales,less “explanation”
2 new physics adds natural explanation of scales by dynamics orsymmetry
3 scales and parameters are explained by unknown quantumgravity
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Strong interactions: dynamic scale generationMassless QCD has no free parameters:
hadronic bound states are dynamical generated
scale of the masses not a free parameter of the theory
theory explains, not parametrizes, the masses
generates naturally large scale separationmassless pions (Goldstone bosons),massive bound states
[B. Lucini]
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Specific extensions of the Standard Model
Composite or Technicolour models:
new strong dynamics beyond the standard model
Higgs generated as bound state, natural due to scale ofadditional strong interactions
Supersymmetry:
natural explanation by symmetry
Higgs mass corrections canceled by fermionic partners
interesting theoretical concept
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The old idea of Technicolour
QCD + weak interactions without Higgs
QCD corrections to W and Z propagators
dominant contribution in massless limit: pion propagator
effectively generates mass for W and Z bosons
QCD
W W W π W m2W = 1
4gf2π
W mass correction by pion decay constant [Weinberg ’76],[Susskind ’79]
SU(NTC )× SU(3)× SU(2)L × UY
Two TC flavours: π±T ,π0T become longitudinal part of W , Z
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The new idea of Technicolour or Composite models
Problems:
Higgs particle observed
Higgs generates fermion masses via Yukawa interactions
Requirements:
need light Higgs particle, large scale separation to other states
need effective interactions with via Extended Technicolour
Approaches:
1 Walking Technicolour approach
2 Composite models
3 Partial compositeness
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The challenges for the constructions of Technicolour orComposite models
You have to explain:
fermion mass generation
electroweak precision data
flavour
Higgs mass
. . .
Since everything should be naturally generated by stronginteractions you have to explain everything at once . . . May be thisis not a good idea.
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Supersymmetry
only possible non-trivial extension of the space-timesymmmetries (Coleman-Mandula theorem)
supersymmetry: every bosonic particle has fermionic partnerwith same mass
Where are the partner particles?
supersymmetry must be broken at lower scales
without knowledge of breaking mechanism: many possiblebreaking terms and parameters, less predictive
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Strong interactions from lattice simulations
still no analytic method to compute low energy features ofstrongly interacting theories
first principles method: Lattice simulations
Well established method for the investigations of QCD, but can weapply it to other theories?
QCD: we have a good feeling about parameters anduncertainties
BSM: we need to redo and even re-think the analysis
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Why needs Supersymmetry non-perturbative methods?
Two main motivations:
1 Supersymmetric particle physics requires breaking terms basedon an unknown non-perturbative mechanism.⇒ need to understand non-perturbative SUSY
2 Supersymmetry as theoretical concept: (Extended) SUSYsimplifies theoretical analysis and leads to newnon-perturbative approaches.
⇒ need to bridge the gap between “beauty” and
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The challenges for supersymmetry on the lattice
discretize, cutoff in momentum space
(controlled) breaking of space-time symmetries⇒ uncontrolled SUSY breaking
derivative operators replaced by difference operators with noLeibniz rule⇒ breaks SUSY
fermonic doubling problem, Wilson mass term
Nielsen-Ninomiya theorem:No-Go for (naive) lattice chiral symmetry⇒ breaks SUSY
gauge fields represented as link variables⇒ different for fermions and bosons
[G.B., S. Catterall, arXiv:1603.04478]
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The challenges for supersymmetry on the lattice
discretize, cutoff in momentum space
(controlled) breaking of space-time symmetries⇒ uncontrolled SUSY breaking
derivative operators replaced by difference operators with noLeibniz rule⇒ breaks SUSY
fermonic doubling problem, Wilson mass term
Nielsen-Ninomiya theorem:No-Go for (naive) lattice chiral symmetry⇒ breaks SUSY
gauge fields represented as link variables⇒ different for fermions and bosons
[G.B., S. Catterall, arXiv:1603.04478]
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N =1 supersymmetric Yang-Mills theory
Supersymmetric Yang-Mills theory:
L = Tr
[−1
4FµνF
µν +i
2λ /Dλ−mg
2λλ
]
supersymmetric counterpart of Yang-Mills theory;but in several respects similar to QCD
λ Majorana fermion in the adjoint representation
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Supersymmetric Yang-Mills theory:Symmetries
SUSY
gluino mass term mg ⇒ soft SUSY breaking
UR(1) symmetry, “chiral symmetry”: λ→ e−iθγ5λ
UR(1) anomaly: θ = kπNc
, UR(1)→ Z2Nc
UR(1) spontaneous breaking: Z2Nc
〈λλ〉6=0→ Z2
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Supersymmetric Yang-Mills theory:effective actions
symmetries + confinement → low energy effective theory
low energy effective actions:1. multiplet1:mesons : a− f0 and a− η′fermionic gluino-glue2. multiplet2:glueballs: 0++ and 0−+
fermionic gluino-glue
Supersymmetry
All particles of a multi-plet must have the samemass.
1[Veneziano, Yankielowicz, Phys.Lett.B113 (1982)]
2[Farrar, Gabadadze, Schwetz, Phys.Rev. D58 (1998)]
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Supersymmetric Yang-Mills theory on the latticeLattice action:
SL = β∑P
(1− 1
Nc<UP
)+
1
2
∑xy
λx (Dw (mg ))xy λy
Wilson fermions:
Dw = 1− κ4∑
µ=1
[(1− γµ)α,βTµ + (1 + γµ)α,βT
†µ
]gauge invariant transport: Tµλ(x) = Vµλ(x + µ);
κ =1
2(mg + 4)
links in adjoint representation: (Vµ)ab = 2Tr[U†µT aUµTb]
of SU(2), SU(3)
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Recovering symmetry
Fine-tuning (Veneziano, Curci)1:
chiral limit = SUSY limit +O(a), obtained at critical κ(mg )
Wilson fermions:
explicit breaking of symmetries: chiral Sym. (UR(1)), SUSY
restored in chiral continuum limit
practical determination of critical κ:
limit of zero mass of adjoint pion (a− π)
⇒ definition of gluino mass: ∝ (ma−π)2
1[Veneziano, Curci, Nucl.Phys.B292 (1987)]
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Supersymmetric Yang-Mills theory: Scale and massdetermination
scale setting:parameters: r0, w0
can be used for comparison to Yang-Mills theory, QCD
meson states (singlet)a–f0 (λλ), a–η′ (λγ5λ)methods for disconnected contributions [PoS LATTICE 2011]
glueballs
spin-1/2 state (gluino-glue)∑µ,ν
σµνtr [Fµνλ]
Technical challenge: Unfortunately all states are quite challengingto determine!
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The particle masses ofN = 1 SU(3) supersymmetric Yang-Mills theory
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
am
(amπ)2
gluino-glueglueball 0++
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
am
(amπ)2
glueball 0++
a-η′a-f0
situation similar to the SU(2) case with comparable latticespacing
indication for a multiplet formation and restored SUSY inchiral limit
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The status of the project
verification of VC scenario: chiral symmetry breaking andsupersymmetric Ward identities
multiplet formation found in the continuum limit of SU(2)SYM [JHEP 1603, 080 (2016)]
SYM thermodynamics: coincidence of chiral anddeconfinement transition [JHEP 1411, 049 (2014)]
SYM compactification: Witten index and absence ofdeconfinement [JHEP 1412, 133 (2014)]
first indication of multiplet formation in the excited states
first SU(3) results with clover improved Wilson fermions
SU(3) very similar to SU(2) case: Multiplet formation.
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Walking Technicolour and the conformal windowWalking Technicolour scenario:
near conformal running of the gauge coupling toaccommodate fermion mass generation and absence of FCNCapproximate conformal symmetry might also lead to naturallight scalar particle (Higgs)
Interesting general question:
conformal window, strong dynamics different from QCDconformal mass spectrum: M ∼ m1/(1+γm) characterised byconstant mass ratios
α∗ conformal
Nf & 12
QCD
Q
α
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Conformal window for adjoint QCD
Gauge theories in higher representation:
smaller number of fermions needed
here: conformal window for adjointrepresentation
mass anomalous dimension γ∗(Nf )
[Dietrich, Sannino,hep-ph/0611341]
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Adjoint QCDadjoint Nf flavour QCD:
L = Tr
[−1
4FµνF
µν +
Nf∑i
ψi ( /D + m)ψi
]
Dµψ = ∂µψ + ig [Aµ, ψ]
ψ Dirac-Fermion in the adjoint representation
adjoint representation allows Majorana condition ψ = C ψT
⇒ half integer values of Nf : 2Nf Majorana flavours
Chiral symmetry breaking:
Z2Nc × SU(2Nf )→ Z2 × SO(2Nf )
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Particle states and lattice action
Lattice action
Wilson fermion action + stout smearing
tree level Symanzik improved gauge action
some results: clover improvement
Particle states
triplet mesons mPS, mS, mV, mPV
glueball 0++
spin-1/2 mixed fermion-gluon state∑µ,ν
σµνtr [Fµνλ]
singlet mesons ma–f0 , ma–η′
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Nf = 2 AdjQCD, Minimal Walking Technicolour
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.05 0.1 0.15 0.2 0.25 0.3 0.35
M/m
PS
amPCAC
√σ/mPS
mV/mPSmspin− 1
2/mPS
m0++/mPSFπ/mPSmS/mPS
mPV/mPS323 × 64483 × 64643 × 64
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
M/m
PS
amPCAC
√σ/mPS
mV/mPSmspin− 1
2/mPS
m0++/mPSFπ/mPSmS/mPS
mPV/mPS
Expected behaviour of a (near) conformal theory:
constant mass ratioslight scalar (0++)no light Goldstone (mPS)
Well established results: [Debbio, Lucini, Patella, Pica, 2016],[GB, Giudice, Munster, Montvay, Piemonte, 2017]
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Results for Nf = 1 adjoint QCD
0
1
2
3
4
5
6
7
8
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
mState / σ
a mPCAC
2⁺ scalar baryon
2⁻ pseudoscalar baryon
2⁻ vector baryon
Spin-½ state
0
1
2
3
4
5
6
7
8
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
mState / σ
a mPCAC
0⁺ scalar meson
0⁻ pseudoscalar meson
0⁺ axial vector meson
0⁻ vector meson
0⁺ glueball
[Athenodorou, Bennett, GB, Lucini, arXiv:1412.5994]
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Results for Nf = 3/2 adjoint QCD
0
0.5
1
1.5
2
2.5
3
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
M/m
PS
amPCAC
323 × 64243 × 48√σ/mPS
mV/mPSmspin− 1
2/mPS
m0++/mPSmS/mPS
mPV/mPS
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.06 0.08 0.1 0.12 0.14 0.16 0.18
M/m
PS
amPCAC
323 × 64243 × 48483 × 96
Nf Dirac fermions → 2Nf Majorana fermions
with our experience of supersymmetric Yang-Mills theory, wecan simulate half integer fermion numbers
requires the Pfaffian sign
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Pfaffian in Nf = 3/2 adjoint QCD
−4
−3
−2
−1
0
1
2
3
4
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
N1.4
sℑλ
N1.4s ℜλ
even at the critical parameters: no sign fluctuations of Pfaffian
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Mass anomalous dimension for Minimal WalkingTechnicolour: Methods
Methods for determination of γ∗:scaling of mass spectrummode number (integrated spectral density of D†D )
Methods for mode number determination:
Chebyshev expansion of the spectral densityconsistency with [Giusti, Luscher, 0812.3638] checked
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2
-0.010
0.010.020.030.040.050.060.070.080.090.1
0.10.150.20.250.30.350.40.450.5
Chebyshev expansiondiagonalisation
projection
-16
-14
-12
-10
-8
-6
-4
-2 -1.5 -1 -0.5 0
Chebyshev expansionprojection
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Mass anomalous dimension for Nf = 3/2
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
γ
Ωmin
κ = 0.1340κ = 0.1320κ = 0.1322
ν(Ω) = 2
∫ √Ω2−m2
0ρH(λ)dλ ; ν(Ω) = ν0 + a1(Ω2 − a2
2)2
1+γ
mass anomalous dimension around 0.38(2)
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Mass anomalous dimension for Nf = 3/2
0
5
10
15
20
25
30
3 4 5 6 7 8 9 10
LM
(L/a)(amPCAC)1/(1+0.5)
323 × 64243 × 48163 × 32LmPSL√σ
LmV
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10
LM
(L/a)(amPCAC)1/(1+0.38)
323 × 64243 × 48163 × 32483 × 96LmPSL√σ
LmV
check with the hyperscaling of the spectrum
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Comparison with Nf = 1 and Nf = 1/2
Theory scalar particle γ∗ small β γ∗ larger βNf = 1/2 SYM part of multiplet – –Nf = 1 adj QCD light 0.92(1) 0.75(4)∗
Nf = 3/2 adj QCD light 0.50(5)∗ 0.38(2)∗
Nf = 2 adj QCD light 0.376(3) 0.274(10)(∗ preliminary)
remnant β dependence: γ∗ not real IR fixed point values
final results require inclusion of scaling corrections
investigation of (near) conformal theory requires carefulconsideration of lattice artefacts and finite size effects
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What can we learn for realistic models?
Next to Minimal Walking Technicolour: Nf = 2 in sextetrepresentation [Bergner, Ryttov, Sannino, 1510.01763]
conformal window for adjoint fermions approximatelyindependent of Nc
large Nc up to factor 2 equivalence between symmetric,adjoint, and antisymmetric representation
small Nc = 2 equivalence between symmetric and adjointrepresentation
conformal behaviour of Nf = 1 indicates conformality ofNMWT
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Towards more realistic theories: Ultra Minimal WalkingTechnicolour
mass anomalous dimension too small in MWT to be a realisticcandidate
Nf = 1 has large mass anomalous dimension, but not therequired particle content
Nf = 1 in adjoint + Nf = 2 in fundamental representation ofSU(2) has been conjectured to be ideal candidate (UMWT)
Nf = 1/2 adjoint + Nf = 2 in fundamental might also beclose enough to conformality
interesting also for further investigations: SQCD withoutscalar fields
test of mixed representation setup for partial compositenessmodels
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First investigations in mixed representation setup
difficult tuning: two bare mass parameters have to be tuned
one-loop clover improved action with Wilson fermions infundamental and adjoint representations
investigate the influence of the adjoint fermions onfundamental sector
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Conclusions and outlook
interesting strongly interacting extensions of the StandardModel with dynamics different from QCD
close connection between the investigations of conformal, ornear conformal models and supersymmetric theories
N = 1 SU(3) supersymmetric Yang-Mills theory showssupersymmetric particle spectrum
interesting new investigations: adjoint+fundamental SU(2),adjoint Nf = 1 coupled to scalar
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