N = 1 Supersymmetric SU(3) Gauge Theory - Towards simulations of Super-QCD Bj¨ornH. Wellegehausen with Andreas Wipf Theoretisch-Physikalisches Institut, FSU Jena 36th International Symposium on Lattice Field Theory, Lansing, Michigan, USA, 27.07.2018 Introduction 2 / 22 Supersymmetric QCD . describes the strong interaction in the MSSM. models the interaction of gluons and quarks with their superpartners, the gluinos and squarks. reduces to N = 1 SYM theory for infinitely heavy quarks and squarks. Interesting non-perturbative aspects are . the mass spectrum and the formation of supermultiplets. the phase diagram at finite temperature. the (Nc; Nf) phase diagram of the massless theory. Supersymmetry is broken by the lattice regularization ) Fine-tune set of operators to restore susy in the continuum limit Outline 3 / 22 1 Supersymmetric QCD in the continuum 2 One-loop effective potential 3 (Preliminary) Lattice results Supersymmetric QCD in the continuum Supersymmetric QCD in the continuum 4 / 22 N = 1 SU(N) SQCD 1 i L = tr − F F µν + λγ¯ µD λ + ¯ (iγµD + m) 4 µν 2 µ µ 1 2 p −D φyD φ − m2φyφ − g 2 φyT σ φ − i 2g φyλ¯P − ¯P¯λφ µ µ 2 3 φ+ P+ with φ = , P = , P¯ = γ0Pγ0 and SU(N) generators T φ− P− Gluons, real vector field Aµ in the adjoint representation (A) Gluinos, Majorana fermions λ in (A) Nf Quarks, Dirac fermions in the fundamental representation (F) 2Nf Squarks, complex scalars φ in the (F) Quark-Squark-Gluino Yukawa interaction SQCD =SYM+QCD+ Scalars+ interaction Supersymmetric QCD in the continuum - Symmetries 5 / 22 Preserved symmetries on the lattice Gauge symmetry Parity: x = (t; x) ! x 0 = (t; −x) 0 0 (x) ! γ0 (x ) ; λ(x) ! γ0λ(x ) and φ±(x) ! −φ∓(x): Baryon number conservation: iα iα U(1)B : ! e ; φ ! e φ ± - exchange symmetry Restrict number of operators that need to be fine-tuned Supersymmetric QCD in the continuum - Symmetries 6 / 22 Explicitly broken on the lattice Chiral symmetry: iαγ iασ ! e 5 ; φ ! e 3 φ anomaly spont. U(1)A −! Z2Nf −! Z2 R symmetry (also in N = 1 SYM): −iαγ iαγ ! e 5 ; λ ! e 5 λ anomaly spont. U(1)R −! Z2(Nc−Nf) −! Z2 anomaly free subgroup anomaly U(1)A ⊗ U(1)R −! U(1)AF Supersymmetric QCD in the continuum - Ward identities 7 / 22 Supersymmetry (N = 1 SYM): p p p µ δφ+ = − 2¯P ; δ = i 2(Dµφ)Pγ − 2mPσ1φ ; α µ α α µν α y α δAµ =−iγ¯ λ ;δ λ = −iσ Fµν − 2igγ5 φ T σ3φ Supersymmetric Ward identities 0 = QO¯ + OQ¯S 1 3 i 3 − tr F F µν = tr λγ¯ Dµλ SYM= N2 − 1 4 µν 8 2 µ 2 c 2 1 − tr F F µν + D φyDµφ + m2φyφ = N2 − 1 + 2N N 3 4 µν µ c c f Konishi anomaly 2 y λλ¯ = 32π m φ σ3φ : Supersymmetric QCD in the continuum - Relevant operators 8 / 22 Fine-tuning need to fine-tune a set of operators O to restore susy in the continuum limit N = 1 SYM: only gluino mass λλ¯ SQCD: all relevant (marginal) operators O with [O] ≤ d = 4 Mass dimension of constituent fields: 3 [A] = [φ] = 1 ; [ ] = [λ] = 2 + quadratic, cubic, and quartic interactions Supersymmetric QCD in the continuum - Relevant operators 9 / 22 quadratic interactions SU(N) gauge invariance and baryon number conservation: F¯ ⊗ F and A ⊗ A Quark and gluino masses ¯ Γ and λ¯ Γ λ with Γ = f1; γ5g 2 squark masses y M1 = tr Φ and M2 = tr (Φσ1) with Φrs = φr φs λγ¯ 5λ breaks parity: Twist can be used to improve the extrapolation to the chiral limit ) next talk by Marc Steinhauser Supersymmetric QCD in the continuum - Relevant operators 10 / 22 cubic interactions SU(N) gauge invariance and baryon number conservation: F¯ ⊗ A ⊗ F only operator compatible with all symmetries is i φyλ¯ P − ¯ P¯ λ φ : appears already in the Lagrange function coupling can be absorbed in a rescaling of the scalar field Supersymmetric QCD in the continuum - Relevant operators 11 / 22 quartic interactions only scalar fields possible SU(N) gauge invariance and baryon number conservation: 1 ⊗ 1 ; A ⊗ A ; F ⊗ F¯ ; R ⊗ R¯ use SU(N) Fierz identities to reduce this 4 types to 1 ⊗ 1 5 independent operators for Nf = 1 2 2 2 2 V1 =Φ++ + Φ−− ; V2 = Φ+− + Φ−+ ; V3 = Φ++Φ−− ; V4 =Φ+−Φ−+ ; V5 = (Φ+− + Φ−+) (Φ++ + Φ−−) : V2 and V5 break chiral symmetry Supersymmetric QCD in the continuum 12 / 22 General Euclidean Lagrange function 1 1 L = tr F 2 + Z D φyD φ + Z m2 M + Z 2 λ V g 2 4 µν φ µ µ φ i i φ i i 1 p + tr λ¯ (γ D − m ) λ + ¯ (γ D − m ) + i 2 φyλ¯ P − ¯ P¯ λ φ 2 µ µ g µ µ q 9 fine-tuning parameters for Nf = 1 and fixed quark mass mq bare gluino mass mg 2 squark masses mi squark wave function renormalization Zφ (Yukawa interaction) 5 squark quartic couplings λi Supersymmetric continuum action: λλ¯ ¯ Zφ M1 M2 V1 V2 V3 V4 V5 2 0 m 1 m 0 (Nc − 1)=Nc 0 1=Nc −1 0 One-loop effective potential One-loop effective potential 13 / 22 Fine-tuning of squark potential guided by lattice perturbation theory Integration over fermions 1 1 S = S + S − tr ln D= − m − tr ln D= − tr ln 1 − ∆ Y ∆ Y¯ y Gauge Squark 2 λ 2 λ ξ Calculate 1-loop effective squark potential 2 ! 2 1-loop 1 δ S['] 1 δ S[φ] Veff (φ) = V (φ) + tr ln − tr ln 2 δ'(x)δ'(y) '=φ 2 δΨ(x)δΨ(y) | {z } | {z } Bosonic contribution Fermionic contribution One-loop effective potential - Continuum result 14 / 22 Continuum VB(p) =CF (2∆φ + 6∆G ) M1 2 2 2 2 2 2 + b1(∆φ; ∆G ; Nc) V1 + b3(∆φ; ∆G ; Nc) V3 + b4(∆φ; ∆G ; Nc) V4 VF(p) =−8CF∆φ M1 2 2 2 2 2 2 + f1(∆φ; ∆G ; m; Nc) V1 + f3(∆φ; ∆G ; m; Nc) V3 + f4(∆φ; ∆G ; m; Nc) V4 1 1 Propagators ∆G = p2 and ∆φ = p2+m2 Only operators of the tree-level potential appear at 1-loop: M1 ; V1 ; V3 ; V4 Quadratic divergences cancel as expected Quartic couplings bi and fi contain logarithmic divergences One-loop effective potential - Lattice result 15 / 22 Lattice Wilson mass and discrete lattice momenta break supersymmetry Replace continuum propagators by lattice propagators All operators are generated at 1-loop continuum X lattice X ^ Veff = λi (k) Oi and Veff = λi (k) Oi k k Improved lattice action SImproved(m; g) = S(m; gjmq; mg; Zφ) + Vcounter(m) 7 X X Vcounter(m) = ∆λi (m; V ; Nc) Oi with ∆λi = (λi (k) − λ^i (k)) i=1 k One-loop effective potential - Lattice result 16 / 22 L M1 M2 V1 V2 V3 V4 V5 8 -1.3793 -0.63618 0.30573 -0.05689 0.25293 0.21088 0.07710 16 -1.3785 -0.63665 0.31012 -0.05722 0.25888 0.22181 0.08316 32 -1.3784 -0.63666 0.31081 -0.05723 0.25983 0.22434 0.08436 64 -1.3784 -0.63666 0.31095 -0.05723 0.26003 0.22495 0.08464 128 -1.3784 -0.63666 0.31099 -0.05723 0.26008 0.22510 0.08471 256 -1.3784 -0.63666 0.31100 -0.05723 0.26009 0.22514 0.08473 (Preliminary) Lattice results (Preliminary) Lattice results - Setup 17 / 22 Lattice setup and phase of the Pfaffian Wilson gauge action with β = 6:3 ::: 9:9, m = 0:5, mq = −0:4 and volume V = 83 × 16 Wilson fermions / rational-HMC algorithm 1.0000 1.0001 0.9998 1.0000 0.9996 0.9999 0.9994 0.9998 0.9997 E E 0.9992 φ φ i i 0.9996 e e 0.9990 0.9995 Re Re 0.9988 D D 0.9994 4 0.9986 V = 3 0.9993 4 0.9984 V = 4 mg = −1:14 4 0.9992 V = 5 mg = −1:18 0.9982 4 0.9991 V = 6 mg = −1:22 0.9980 0.9990 -1.28 -1.26 -1.24 -1.22 -1.20 -1.18 -1.16 -1.14 -1.12 0 200 400 600 800 1000 1200 1400 mg V Extrapolation: 0:996 ::: 0:998 @ 83 × 16 and 0:94 ::: 0:97 @ 163 × 32 (Preliminary) Lattice results - Setup 18 / 22 Fine-tuning of mg adjoint pion fundamental pion 0.8 0.92 unimproved, m = 0:5 0.90 0.7 improved, m = 0:5 0.88 improved, m = 0:3 0.6 0.86 0.84 0.5 0.82 m m 0.4 0.80 0.78 0.3 0.76 unimproved, m = 0:5 0.74 0.2 improved, m = 0:5 0.72 improved, m = 0:3 0.1 0.70 -1.16-1.14-1.12-1.10-1.08-1.06-1.04-1.02-1.00-0.98 -1.16-1.14-1.12-1.10-1.08-1.06-1.04-1.02-1.00-0.98 mg mg we use the adjoint pion mass to fine-tune mg fundamental pion mass almost independent of gluino mass and squark mass (Preliminary) Lattice results - Setup 19 / 22 SQCD compared to SYM -0.60 -0.70 -0.80 -0.90 mg -1.00 -1.10 -1.20 SQCD SYM -1.30 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 β massless gluino critical lines of SQCD and SYM coincide almost independent of squark /quark mass and scalar potential (Preliminary) Lattice results - Setup 20 / 22 Unimproved vs improved Ward identity 0.07 0.07 0.06 0.06 0.05 0.05 W W 0.04 0.04 0.03 0.03 0.02 0.02 -1.30 -1.20 -1.10 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -1.30 -1.20 -1.10 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 mg mg 2 hS i + D φyDµφ + m2φyφ = N2 − 1 + 2N N = 14 3 W µ c c f (Preliminary) Lattice results - Setup 20 / 22 Unimproved vs improved Ward identity 0.065 unimproved 0.060 improved 0.055 0.050 0.045 W 0.040 0.035 0.030 0.025 0.020 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 β b Fit to W (β) = W1 + aβ unimproved improved W1 = 0:008(3) and W1 = −0:003(11) Conclusions and Outlook 21 / 22 Conclusions and Outlook No severe sign problem in lattice simulations One-loop improved lattice action Preliminary lattice results for Ward identities, the chiral critical line, fundamental / adjoint pion mass Fine-tuning of the Yukawa interaction Gluino-glue and Quark-Squark bound states Literature M.