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 + ψ¯ (iγµD + m) ψ 4 µν 2 µ µ 1 2 √ −D φ†D φ − m2φ†φ − g 2 φ†T σ φ
− i 2g φ†λ¯Pψ − ψ¯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 ψ 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 ψ(x) → γ0ψ(x ) , λ(x) → γ0λ(x ) and φ±(x) → −φ∓(x).

Baryon number conservation:

iα iα U(1)B : ψ → e ψ , 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: ψ → eiαγ5 ψ , φ → eiασ3 φ

anomaly spont. U(1)A −→ Z2Nf −→ Z2 R symmetry (also in N = 1 SYM):

ψ → e−iαγ5 ψ , λ → eiαγ5 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): √ √ √ µ δφ+ = − 2¯Pψ , δψ = i 2(Dµφ)Pγ  − 2mPσ1φ  , α µ α α µν α † 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 φ†Dµφ + m2φ†φ = N2 − 1 + 2N N 3 4 µν µ c c f

Konishi anomaly

2 † 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 , [ψ] = [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

ψ¯ Γ ψ and λ¯ Γ λ with Γ = {1, γ5}

2 squark masses † 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 φ†λ¯ P ψ − ψ¯ 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 φ†D φ + Z m2 M + Z 2 λ V g 2 4 µν φ µ µ φ i i φ i i 1 √ + tr λ¯ (γ D − m ) λ + ψ¯ (γ D − m ) ψ + i 2 φ†λ¯ P ψ − ψ¯ 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¯ ψψ¯ 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¯ †
Gauge Squark ψ 2 λ 2 λ ξ Calculate 1-loop effective squark potential

2 !  2  1-loop 1 δ S[ϕ] 1 δ S[φ] Veff (φ) = V (φ) + tr ln − tr ln 2 δϕ(x)δϕ(y) ϕ=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 ˆ Veff = λi (k) Oi and Veff = λi (k) Oi k k

Improved lattice action

SImproved(m, g) = S(m, g|mq, mg, Zφ) + Vcounter(m)

7 X X Vcounter(m) = ∆λi (m, V , Nc) Oi with ∆λi = (λi (k) − λˆi (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 φ†Dµφ + m2φ†φ = 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) = W∞ + aβ

unimproved improved W∞ = 0.008(3) and W∞ = −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. J. Strassler: QCD, supersymmetric QCD, lattice QCD and string theory: Synthesis on the horizon? Ian Affleck, Michael Dine, and Nathan Seiberg: Dynamical Supersymmetry Breaking in Supersymmetric QCD Joel Giedt: Progress in four-dimensional lattice supersymmetry M. Costa and H. Panagopoulos: Supersymmetric QCD on the Lattice: An Exploratory Study Conclusions and Outlook - Fermion integration 22 / 22

Coupling between Dirac- and Majorana fermions Rewrite fermion Lagrange function

 /    1 ¯ D i Y 1 T CD/ i CY Lf = Ψ ¯ † Ψ = Ψ T Ψ 2 −i Y D/ξ − m 2 −i (CY ) C(D/ξ − m) | {z } M

Y = φˆ†PeT − φˆTPe¯ † , Y¯ † = e∗P¯φˆ − ePφˆ∗ Majorana spinor Ψ and M = −MT ⇒ Pfaffian

Bosonic effective action

1 1 S = S + S − tr ln D/ − m
− tr ln D/ − tr ln 1 − ∆ Y ∆ Y¯ †
eff Gauge Squark ψ 2 λ 2 λ ξ