Lattice simulations and extensions of thestandard model with strong interactions

Georg BergnerFSU Jena

SIFT 2017: Jena 23. 11. 2017

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

2

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

3

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

4

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

5

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]

6

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

7

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

8

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

9

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.

10

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

11

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

12

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

13

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]

14

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]

14

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

15

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

16

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)]

17

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)

18

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)]

19

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!

20

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

21

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.

22

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

23

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]

24

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 )

25

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′

26

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]

27

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]

28

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

29

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

30

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

31

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)

32

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

33

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

34

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

35

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

36

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

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

w0mV

(amπ)2

0

0.2

0.4

0.6

0.8

1

6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

m2 π

1/κ

κfκfκaκa

37

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

38