Simulations of Supersymmetric and Near Conformal Gauge Theories on the Lattice

Simulations of supersymmetric and near conformal gauge theories on the lattice

Georg Bergner FSU Jena

Trento: September 3, 2018 Lattice SUSY SYM Bound states Conf

Supersymmetric and conformal theories

Two diﬀerent approaches to BSM physics. . . 1 supersymmetry: extensions by symmetry 2 compositeness: new strong scale by new strong dynamics . . . lead to similar problems tuning on the lattice and the running coupling realization of conformal symmetry on the lattice non-standard strongly coupled theories with fermions in higher/mixed representations

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Alternative solutions to the Hierarchy problem

Compositeness: new strong dynamics beyond the standard model Higgs generated as bound state, natural due to scale of additional strong interactions Symmetry: natural explanation by symmetry Higgs mass corrections canceled by fermionic partners Both approaches lead also to interesting theoretical concepts that go beyond the phenomenological applications.

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1 Supersymmetry on the lattice

2 Supersymmetric Yang-Mills theory on the lattice

3 The bound state spectrum of SU(3) supersymmetric Yang-Mills theory

4 Towards supersymmetric QCD: near conformal strong dynamics and SUSY theories in collaboration with S. Ali, H. Gerber, P. Giudice, S. Kuberski, C. Lopez, G. Munster,¨ I. Montvay, S. Piemonte, P. Scior

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Why study SUSY on the lattice?

1 BSM physics: Supersymmetric particle physics requires breaking terms based on an unknown non-perturbative mechanism. need to understand non-perturbative SUSY ⇒ 2 Supersymmetry is a general beautiful theoretical concept: (Extended) SUSY simpliﬁes theoretical analysis and leads to new non-perturbative approaches. need to bridge the gap between “beauty” and ⇒

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Lattice simulations of SUSY theories Lattice simulations would be the ideal method to investigate non-perturbative sector of SUSY theories . . .

Theory: next part → Can we deﬁne a lattice SUSY? Can we control SUSY breaking?

Practical Simulations: example SYM → SUSY theories have nice properties, but require to rework numerical methods

. . . but are challenging from theoretical and practical point of view.

[G.B., S. Catterall, arXiv:1603.04478]

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Lattice simulations of SUSY theories Lattice simulations would be the ideal method to investigate non-perturbative sector of SUSY theories . . .

Theory: next part → Can we deﬁne a lattice SUSY? Can we control SUSY breaking?

Practical Simulations: example SYM → SUSY theories have nice properties, but require to rework numerical methods

. . . but are challenging from theoretical and practical point of view.

[G.B., S. Catterall, arXiv:1603.04478]

6/50 “The lattice is the only valid non-perturbative deﬁnition of a QFT and it can not be combined with SUSY. Therefore SUSY can not exist!” (Lattice theorist)

Lattice SUSY SYM Bound states Conf

SUSY breaking and the Leibniz rule on the lattice

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SUSY breaking and the Leibniz rule on the lattice

Like Nielsen-Ninomiya theorem: locality contradicts with SUSY On the lattice: There is no Leibniz rule for a discrete derivative operator. The action can only be invariant with a non-local derivative and non-local product rule. [GB],[Kato,Sakamoto,So],[Nicolai,Dondi] Further problems: fermonic doubling problem, Wilson mass term gauge ﬁelds represented as link variables “The lattice is the only valid non-perturbative deﬁnition of a QFT and it can not be combined with SUSY. Therefore SUSY can not exist!” (Lattice theorist)

7/50 . . . but we still don’t completely understand her lesson.

Lattice SUSY SYM Bound states Conf

General solution by generalized Ginsparg-Wilson relation? “Mrs. RG, the good physics teacher. . . ” (Peter Hasenfratz)

Symmetry in the continuum (S[(1 + εM˜)ϕ] = S[ϕ]) implies relation for lattice action SL: Generalized Ginsparg-Wilson relation

2 ! ij j δSL 1 ij δSL δSL δ SL M φ = (Mα− ) nm m i nm j i j i δφn δφm δφn − δφmδφn

Φ[M˜ϕ] = MnmΦm[ϕ]

Still open problem how to ﬁnd solutions. [GB, Bruckmann, Pawlowski]

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General solution by generalized Ginsparg-Wilson relation? “Mrs. RG, the good physics teacher. . . ” (Peter Hasenfratz)

Symmetry in the continuum (S[(1 + εM˜)ϕ] = S[ϕ]) implies relation for lattice action SL: Generalized Ginsparg-Wilson relation

2 ! ij j δSL 1 ij δSL δSL δ SL M φ = (Mα− ) nm m i nm j i j i δφn δφm δφn − δφmδφn

Φ[M˜ϕ] = MnmΦm[ϕ]

Still open problem how to ﬁnd solutions. [GB, Bruckmann, Pawlowski]

. . . but we still don’t completely understand her lesson.

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Sketch of solutions

The Ward-idenities of the unimproved Wilson model The Ward-idenities of the improved Wilson model (m = 10, g = 800, N = 21) (m = 10, g = 800, N = 21) 0.0025 0.006 Ward-Identity 1 Ward-Identity 1 0.002 Ward-Identity 2 Ward-Identity 2 0.004 0.0015

0.001 0.002 0.0005 (2) n (2) n R R

, 0 , 0 (1) n (1) n

R -0.0005 R -0.002 -0.001

-0.0015 -0.004 -0.002

-0.0025 -0.006 0 5 10 15 20 0 5 10 15 20 lattice point n lattice point n The Ward-idenities of the full supersymmetric model (m = 10, g = 800, N = 15) 0.006 SLAC unimproved Ward-Identity 1 SLAC improved Ward-Identity 1 only model dependent solutions Ward-Identity 1 0.004 Ward-Identity 2

0.002 partial realization of extended (2) n R , 0 SUSY (1) n R -0.002 non-local actions -0.004

-0.006 otherwise: ﬁne tuning. 0 2 4 6 8 10 12 14 lattice point n

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Super Yang-Mills theory

Supersymmetric Yang-Mills theory: 1 µν i mg = Tr Fµν F + λ¯D/λ λλ¯ L −4 2 − 2

supersymmetric counterpart of Yang-Mills theory; but in several respects similar to QCD λ Majorana fermion in the adjoint representation SUSY transformations: δA = 2iλγ¯ ε, δλ = σ F ε µ − µ − µν µν

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Why study supersymmetric Yang-Mills theory on the lattice ?

1 extension of the standard model gauge part of SUSY models understand non-perturbative sector: check eﬀective actions etc.

2 controlled confinement [Unsal,Yaffe,¨ Poppitz] : compactified SYM: continuity expected small R regime: semiclassical confinement

3 connection to QCD [Armoni,Shifman]: orientifold planar equivalence: SYM QCD Remnants of SYM in QCD ? ↔ comparison with one ﬂavor QCD

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Supersymmetric Yang-Mills theory: Symmetries

SUSY

gluino mass term mg soft SUSY breaking ⇒

iθγ5 UR (1) symmetry, “chiral symmetry”: λ e− λ kπ → UR (1) anomaly: θ = , UR (1) Z2Nc Nc → λλ¯ =0 h i6 UR (1) spontaneous breaking: Z2Nc Z2 →

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Supersymmetric Yang-Mills theory on the lattice Lattice action: X 1 1 X ¯ L = β 1 UP + λx (Dw (mg ))xy λy S − Nc < 2 P xy

Wilson fermions: 4 X h i Dw = 1 κ (1 γ ) T + (1 + γ ) T † + clover − − µ α,β µ µ α,β µ µ=1

gauge invariant transport: Tµλ(x) = Vµλ(x +µ ˆ); 1 κ = 2(mg + 4) a b links in adjoint representation: (Vµ)ab = 2Tr[Uµ†T UµT ] of SU(2), SU(3)

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Lattice SYM: Symmetries

Wilson fermions:

explicit breaking of symmetries: chiral Sym. (UR (1)), SUSY ﬁne tuning: add counterterms to action tune coeﬃcients to obtain signal of restored symmetry special case of SYM: 1 tuning of mg enough to recover chiral symmetry same tuning enough to recover supersymmetry 2

1 [Bochicchio et al., Nucl.Phys.B262 (1985)] 2 [Veneziano, Curci, Nucl.Phys.B292 (1987)]

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Recovering symmetry

Fine-tuning: chiral limit = SUSY limit+ O(a), obtained at critical κ(mg )

no ﬁne tuning with Ginsparg-Wilson fermions (overlap/domainwall) fermions3; but too expensive practical determination of critical κ: limit of zero mass of adjoint pion (a π) 2− deﬁnition of gluino mass: (ma π) ⇒ ∝ − cross checked with SUSY Ward identities

3 [Fleming, Kogut, Vranas, Phys. Rev. D 64 (2001)], [Endres, Phys. Rev. D 79 (2009)], [JLQCD, PoS Lattice 2011]

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The sign problem in supersymmetric Yang-Mills Majorana fermions: Z 1 R λ¯Dλ n λe− 2 = Pf(CD) =( 1) √det D D − n = number of degenerate real negative eigenvalue pairs 0.02

0.015

0.01

0.005 no sign problem λ 0 ℑ @ current 0.005 − parameters 0.01 − 0.015 − 0.02 − 0.006 0.004 0.002 0 0.002 0.004 0.006 0.008 0.01 − − − λ ℜ 16/50 Supersymmetry Particles must have same mass.

Lattice SUSY SYM Bound states Conf

Low energy eﬀective theory

multiplet1 multiplet2 ++ scalar meson a f0 glueball 0 − + pseudoscalar meson a η0 glueball 0− fermion gluino-glue− gluino-glue

confinement: colourless bound states symmetries + confinement low energy effective theory → glueballs, gluino-glueballs, gluinoballs (mesons) build from chiral multiplet type

1 [Veneziano, Yankielowicz, Phys.Lett.B113 (1982)] 2 [Farrar, Gabadadze, Schwetz, Phys.Rev. D58 (1998)]

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Low energy eﬀective theory

multiplet1 multiplet2 ++ Supersymmetry scalar meson a f0 glueball 0 − + Particles must ha- pseudoscalar meson a η0 glueball 0− fermion gluino-glue− gluino-glue ve same mass.

confinement: colourless bound states symmetries + confinement low energy effective theory → glueballs, gluino-glueballs, gluinoballs (mesons) build from chiral multiplet type

1 [Veneziano, Yankielowicz, Phys.Lett.B113 (1982)] 2 [Farrar, Gabadadze, Schwetz, Phys.Rev. D58 (1998)]

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Bound states in supersymmetric Yang-Mills theory

like in YM and QCD: glueball bound states of gluons meson states (like ﬂavour singlet mesons in QCD)

a–f0 : λλ¯ ; a–η0 : λγ¯ 5λ

gluino-glue spin-1/2 state

X µν σµνtr [F λ] µ,ν

Quite challenging to get good signal for the correlators of these operators. Mass determined from exponential decay of the correlator.

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The status of the project Advanced methods of lattice QCD required:

disconnected contributions [LATTICE2011]

eigenvalue measurements [GB,Wuilloud] variational methods (including mixing of glueball and meson operators) [LATTICE2017] SU(2) SYM: multiplet formation found in the continuum limit of SU(2) SYM [JHEP 1603, 080 (2016)] SU(3) SYM: adjoint representation much more demanding than fundamental one (limited to small lattice sizes)

ﬁrst SU(3) simulations [LATTICE99,LATTICE2016,LATTICE2017]

results presented here: [arXiv:1801.08062]

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The fermion-boson mass degeneracy

1.6 glueball 0++ 1.4 gluino-glue

1.2

0.8 am

0.6

0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 (amπ)

gluino-glue and glueballs become degenerate in the chiral limit (lattice size 163 32, β = 5.5) ×

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The mesonic states and complete multiplet

glueball 0++ 1 1 gluino-glue a-f0 a-η 0.8 0.8

0.6 0.6 am am

0.4 0.4

glueball 0++ 0.2 gluino-glue 0.2 β = 5.5 a–f0 β = 5.6 a–η0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.05 0.1 0.15 0.2 0.25 0.3 2 2 (amπ) (amπ)

within errors: degeneracy of SUSY multiplet at two diﬀerent lattice spacings

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Coarser lattices: gap in the particle spectrum

1.6 1.4 glueball 0++ glueball 0++ gluino-glue gluino-glue 1.4 1.2

1.2 1 1 0.8 0.8 am am 0.6 0.6 0.4 0.4

0.2 0.2

0 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 2 2 (amπ) (amπ) complete continuum extrapolations: coming soon

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Going beyond = 1 SYM N

General tuning approach: O(a) SUSY breaking on the lattice radiative corrections lead to relevant breaking, compensated by counterterms required tuning: all operators with dimension less than four simpliﬁed approach: assume Ginsparg-Wilson fermions that preserve the R-symmetry

[J. Giedt,Int.J.Mod.Phys. A24 (2009)] important additional problem: conformal theories (S-duality, ⇒ = 4 SYM . . . ) N

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Some history of composite Higgs and Technicolour

The attractive idea of Technicolour natural introduction of a new dynamical generated scale by additional strong interactions the failure and the recovery plain Technicolour can not explain the large diﬀerence between the suppressed FCNC and fermion mass generating operators cure might be due to a walking behaviour of the running coupling

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Walking Technicolour and the conformal window Walking Technicolour scenario: near conformal running of the gauge coupling to accommodate fermion mass generation and absence of FCNC approximate conformal symmetry might also lead to natural light scalar particle (Higgs) Interesting general question: conformal window, strong dynamics diﬀerent from QCD conformal mass spectrum: M m1/(1+γm) characterised by constant mass ratios ∼ α QCD

α∗ conformal

Nf & 12

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Conformal window for adjoint QCD

Gauge theories in higher representation: smaller number of fermions needed here: conformal window for adjoint representation

mass anomalous dimension γ (Nf ) ∗ [Dietrich, Sannino, hep-ph/0611341]

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Adjoint QCD adjoint Nf ﬂavour QCD:

" Nf # 1 µν X = Tr Fµν F + ψ¯i (D/ + m)ψi L −4 i

Dµψ = ∂µψ + ig[Aµ, ψ]

ψ Dirac-Fermion in the adjoint representation adjoint representation allows Majorana condition ψ = Cψ¯T

half integer values of Nf : 2Nf Majorana ﬂavours ⇒ Chiral symmetry breaking:

Z2N SU(2Nf ) Z2 SO(2Nf ) c × → ×

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Nf = 2 AdjQCD, Minimal Walking Technicolour 1.6 1.4 √σ/mPS √σ/mPS m /m m /m 1.4 V PS V PS mspin 1 /mPS 1.2 mspin 1 /mPS − 2 − 2 m0++ /mPS m0++ /mPS 1.2 F /m F /m π PS 1 π PS mS/mPS mS/mPS 1 mPV/mPS mPV/mPS 323 64 0.8 PS 3 × PS 0.8 48 64 643 × 64

M/m × M/m 0.6 0.6 0.4 0.4

0.2 0.2

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26

amPCAC amPCAC Expected behaviour of a (near) conformal theory: constant mass ratios light scalar (0++) no light Goldstone (mPS) Well established results: [Debbio, Lucini, Patella, Pica, 2016],[Catterall,Sannino,2007]

[Catterall,Del Debbio,Giedt, Keegan,2012],[GB, Giudice, Munster,¨ Montvay, Piemonte, 2017]

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Results for Nf = 3/2 adjoint QCD

3 3 3 32 64 1.6 32 64 243 × 48 243 × 48 √σ/m× 483 × 96 2.5 PS 1.4 × mV/mPS mspin 1 /mPS − 2 1.2 2 m0++ /mPS mS/mPS mPV/mPS 1 PS PS 1.5 0.8 M/m M/m 1 0.6 0.4 0.5 0.2

0 0 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.06 0.08 0.1 0.12 0.14 0.16 0.18

amPCAC amPCAC

Nf Dirac fermions 2Nf Majorana fermions → with our experience of supersymmetric Yang-Mills theory, we can simulate half integer fermion numbers requires the Pfaﬃan sign

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Results for Nf = 1 adjoint QCD

2⁺ scalar baryon 2⁻ vector baryon 0⁺ scalar meson 0⁻ vector meson 2⁻ pseudoscalar baryon Spin-½ state 0⁻ pseudoscalar meson 0⁺ glueball 0⁺ axial vector meson 8 8

7 7

6 6

/ � σ 5 / � σ 5 State State m 4 m 4

3 3

2 2

1 1

0 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

a mPCAC a mPCAC

[Athenodorou, Bennett, GB, Lucini, arXiv:1412.5994]

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Comparison with of adjoint QCD with diﬀerent Nf

Theory scalar particle γ∗ small β γ∗ larger β Nf = 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 careful consideration of lattice artefacts and finite size effects

31/50 Lattice SUSY SYM Bound states Conf

Towards more realistic theories: Ultra Minimal Walking Technicolour

mass anomalous dimension too small in MWT to be a realistic candidate

Nf = 1 has large mass anomalous dimension, but not the required particle content

Nf = 1 in adjoint + Nf = 2 in fundamental representation of SU(2) has been conjectured to be ideal candidate (UMWT)

Nf = 1/2 adjoint + Nf = 2 in fundamental might also be close enough to conformality

32/50 Lattice SUSY SYM Bound states Conf

Extensions towards SUSY theories

Relation to SUSY theories:

Nf = 1 adjoint QCD corresponds to = 2 SYM without scalars N

Nf = 1/2 adjoint + Nf = 2 fundamental corresponds to SQCD without scalars Ongoing work: test mixed representation setup with UMWT extend studies with scalars towards SUSY theories

33/50 Lattice SUSY SYM Bound states Conf

UMWT: Cross check in pure Nf = 2 SU(2) fundamental theory 3.5 reference

2.5 v

m 2 0 w

1.5

0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2 (amπ) reasonable agreement with recent (continuum extrapolated) results [Arthur,Drach,Hansen,Hietanen,Pica,Sannino] larger β to avoid possible bulk transition

(SU(2) Nf = 1 adjoint) 34/50 Lattice SUSY SYM Bound states Conf

UMWT: First investigations in mixed representation setup: tuning

Fixed κadj = 0.1600 1.6

1.4

1.2

1 adj ) 0.8 2 π m ( 0.6

0.4

0.2

0 7 7.5 8 8.5 9 9.5 10

1/κfund one-loop improved Wilson clover fermions: tuning of fundamental and adjoint not independent

35/50 Lattice SUSY SYM Bound states Conf

UMWT: First investigations in mixed representation setup: tuning

Fixed κfund = 0.1350 0.7

0.6

0.5

0.4 fund ) 2 π

m 0.3 (

0.2

0.1

0 6 6.5 7 7.5 8 8.5 9 9.5 10

1/κadj one-loop improved Wilson clover fermions: tuning of fundamental and adjoint not independent

36/50 Lattice SUSY SYM Bound states Conf

UMWT: First investigations in mixed representation setup

1.9 κa = 0 κa = 0.1600 1.8 κa = 0.1620

1.7

1.6

1.5 π

/m 1.4 v m 1.3

1.2

1.1

0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

amπ adjoint ﬂavour drives theory towards near conformal behaviour

37/50 Lattice SUSY SYM Bound states Conf

= 1 SYM and mixed representations: N supersymmetric QCD

add Nc N¯c chiral matter superﬁeld to supersymmetric Yang-Mills⊕ theory

SYM + quarks ψ and squarks Φi with covariant derivatives, mass terms and

¯a a a i√2gλ Φ1†T P+ + Φ2T P ψ − ¯ a a a i√2gψ P T Φ1 + P+T Φ2† λ − − 2 2 g a a Φ†T Φ1 Φ†T Φ2 . 2 1 − 2

38/50 Lattice SUSY SYM Bound states Conf

Why we consider SQCD

natural extension of supersymmetric Yang-Mills theory relation to possible extensions of the standard model earlier studies of lattice formulation: perturbative [Costa, Panagopoulos], tuning [Giedt, Veneziano] SQCD analysis of Seiberg et al.:

Nf < Nc No vacuum

Nf = Nc conﬁnement and chiral symmetry breaking 3 2 Nc < Nf < 3Nc infrared ﬁxed point (duality) Like other SUSY theories beyond = 1 SYM: conformal or near conformal behaviour N

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Why we should better not consider SQCD

large space of tuning parameters [Giedt] (O(10) parameters) just test the mismatch might need formulation with Ginsparg-Wilson fermions still test it with Wilson fermions complex Pfaﬃan related to bosonic symmetry transforming Pf Pf → ∗ not well behaved chiral limit: either near conformal test near conformal scenario in a related theory or unstable vacuum test with Nf = 1 SQCD

40/50 Lattice SUSY SYM Bound states Conf

Why we should better not consider SQCD

40/50 Lattice SUSY SYM Bound states Conf

Nf = 1 SU(2) SQCD vacuum

12 Φ ag d

Φ 10

2 0 20 40 60 80 100 120 140 160 180 HMC the expected instability when going chiral

41/50 Lattice SUSY SYM Bound states Conf

Nf = 1 SU(2) SQCD vacuum

0.2

0.18

0.16

0.14

0.12

0.1 fund κ 0.08

0.06

0.04

0.02

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

κadj constraint phase diagram for the parameter tuning simulations with an O(g 0) SUSY action

42/50 Lattice SUSY SYM Bound states Conf

Conclusions

simulation of supersymmetric theories on the lattice is still in some aspects an open theoretical problem Supersymmetric Yang-Mills theory: theoretical problem is solvable, practical challenges SUSY breaking is under control and formation of chiral multiplet observed for the gauge groups SU(2) and SU(3) interesting non-perturbative physics like the phase diagram can be investigated on the lattice further aspects of the spectrum are currently investigated: mixing of glueballs and gluinoballs, excited states, further bound states

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Conclusions

Beyond SYM: challenging tuning problem at the same time challenging additional problem: non QCD-like behaviour learn from the investigations of conformal window already interesting to study non SUSY related versions of SQCD and = 2 SYM N ongoing work: Yukawa couplings + Scalar potential Requires analysis in a regime where SUSY is restored in SYM (at least 243 48 lattice with unimproved action) ×

44/50 Lattice SUSY SYM Bound states Conf

SU(2) supersymmetric Yang-Mills theory at finite temperature Deconfinement: deconf. above Tc plasma of gluons and gluinos Order parameter: Polyakov loop Chiral phase transitions: chiral above Tc fermion condensate melts and chiral symmetry gets restored order parameter: λλ¯ h i In QCD: quarks add screening effects explicit chiral symmetry breaking both transitions become crossover → In SYM: two independent transitions (at mg = 0)

45/50 Lattice SUSY SYM Bound states Conf

Lattice results SYM at ﬁnite T

< PL > <ΛΛ>S 0.04 0.3 È È 0.03 0.2 0.02

0.1 0.01

0.75 1. 1.25 1.5 1.75 2. T Tc 0.75 1. 1.25 1.5 1.75 2. T Tc second order deconﬁnement transition Tc (SYM) = 0.826(18). Tc (pure Yang-Mills) coincidence of deconﬁnement and chiral transition chiral deconf. Tc = Tc (within current precision) [JHEP 1411 (2014) 049]

46/50 Lattice SUSY SYM Bound states Conf

Compactiﬁed SYM with periodic boundary conditions

0.007 0.7 I II III 0.006 0.6 0.005 0.5 periodic κ = 0.16 0.004 0.4 thermal κ = 0.16 |i P L periodic κ = 0.17 χ P κ = 0.17 h| thermal 0.003 0.3 periodic κ = 0.188 thermal κ = 0.188 0.2 0.002

0.1 0.001

0 0 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 β β

fermion boundary conditions: thermal periodic → at small m (large κ) no signal of deconfinement intermediate masses: two phase transitions (deconfinement + reconfinement) [GB,Piemonte]

47/50 Lattice SUSY SYM Bound states Conf

Phase diagram at ﬁnite temperature/compactiﬁcation

∞ /T ;1

R confined

1 QCDadj Tc

1 thermal b.c. deconfined Y M Tc

thermal and period. b.c. deconﬁned

0 m ∞

change of boundary conditions in compact direction Z(βB ) Z˜(βB ) (Witten index) → Witten index can not have βB dependence: states can only be lifted pairwise continuity in SYM ⇒

48/50 Lattice SUSY SYM Bound states Conf

Pfaﬃan in Nf = 3/2 adjoint QCD

1 λ ℑ 4

. 0 1 s N 1 − 2 − 3 − 4 − 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 − − N 1.4 λ s ℜ even at the critical parameters: no sign ﬂuctuations of Pfaﬃan

49/50 Lattice SUSY SYM Bound states Conf

What can we learn for phenomenological models?

Next to Minimal Walking Technicolour: Nf = 2 SU(3) in sextet representation [Bergner, Ryttov, Sannino, 1510.01763] conformal window for adjoint fermions approximately independent of Nc

large Nc up to factor 2 equivalence between symmetric, adjoint, and antisymmetric representation

small Nc = 2 equivalence between symmetric and adjoint representation

conformal behaviour of Nf = 1 indicates conformality of NMWT

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