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Renormalization Group Flows and Supersymmetry Renormalization Group Flows and Supersymmetry A. Wipf Theoretisch-Physikalisches Institut, FSU Jena What is Quantum Field Theory? Manolo-Fest, Benasque, September 2011 Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 1 / 31 1 Functional renormalization flows 2 Manifest supersymmetric flows for non-gauge models 3 Scale-dependence of super-potential 4 Masses, Phases, Fixed Points, . 5 Exact solution for O(N)-models in large N limit Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 2 / 31 strong-coupling phenomena: spectrum, phase transitions, collective condensations, etc. exact solutions: low dimensions, extended symmetry, integrable models, . strong coupling expansions, large N,... lattice simulations: problems: symmetries, doublers, chiral fermions, sign-problem, . =) wish complement to lattice studies functional renormalization group: phase transitions, condensed matter systems, infrared sector of gauge theories, quantum gravity, renormalizability-proofs, . QFT = solution of functional renormalization group equation Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 3 / 31 Functional renormalization flow Wilson, Wegner-Houghton, Polchinsky, Wetterich generating functional for n−point functions: formally Z Z[j] = Dφ e−S[φ]+(j,φ) ; W [j] = log Z [j] add momentum- and scale-dependent regulator term to S 1 Z dd p 1 ∆S [φ] = φ∗(p)R (p)φ(p) = (φ, R φ) k 2 (2π)d k 2 k =) scale-dependent generating functionals Z −S[φ]+(j,φ)−∆Sk [φ] Zk [j] = Dφ e ; Wk [j] = log Zk [j] Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 4 / 31 Requirements Rk regulates QFT in UV andIR recover full generating functionals in IR: k!0 Rk (p) −! 0 fixed p k!Λ recover classical theory at cutoff Λ: Rk −! 1 =) interpolation between classical and quantum theory IR regularization Rk (p) > 0 for p ! 0 2 2 2 2 e.g. optimized regulator Rk (p) = (k − p ) θ k − p Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 5 / 31 compatible with symmetries flow equation for Wk [j]: Polchinsky equation 1 1 @ W = − tr@ R W (2) − W (1);@ R W (1) k k 2 k k k 2 k k k k 0 Wk (n): n th functional derivative of Wk Legendre transform ) scale dependent effective action Γk ['] = (LWk )['] − ∆Sk ['] Γk not necessarily convex, Γ = Γk!0 convex average field $ source δW [j] δΓ '(x) = k ; k = j(x) − (R ')(x) δj(x) δ'(x) k Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 6 / 31 exact flow equation for Γk [']: Wetterich equation ! 1 @ R @ Γ ['] = tr k k k k 2 (2) Γk ['] + Rk nonlinear functional differential equation, 1-loop structure denominator: IR regularization numerator: @k Rk peaked at p = k ) quantum fluctuations near momentum shell p ≈ k integrate ’down’: collect quantum fluctuations below cutoff Λ nonperturbative, applicable to QFT’s and statistical systems in particular: supersymmetric QFT’s Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 7 / 31 Γk=Λ = S Γk=0 = Γ Theory space Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 8 / 31 Challenges supersymmetry: Poincare algebra, fermionic supercharges i ¯j ij µ i j ij fQα; Qα_ g = 2δ σαα_ Pµ ; fQα; Qβg = 2"αβZ symmetry of renormalization flow =) Ward identities ) mass degeneracy, non-renormalization theorems, . dynamical susy breaking =) phase transitions fixed-point structure of susy theories finite temperature effects here: simple susy Yukawa models I d = 2: infinitely many fixed point solutions: superconformal QFT’s I d = 3: two fixed point solutions; susy breaking, finite T ,... Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 9 / 31 Wess-Zumino models in 2 and 3 dimensions N = 1: real superfield field content: scalar φ, Majorana , auxiliary F off-shell Yukawa model 1 µ i ¯= 1 2 1 00 ¯ 0 LE = 2 @µφ∂ φ + 2 @ − 2 F + 2 W (φ) γ∗ − FW (φ) eliminate dummy F =) on-shell Yukawa model 1 2 i ¯= 1 0 2 1 00 ¯ LE = 2 (@φ) + 2 @ + 2 W (φ)+ 2 W (φ) γ∗ classical model () superpotential W Witten index ) W (φ) ∼ φm: m even: susy always unbroken m odd: susy breaking possible Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 10 / 31 supersymmetric regulator on-shell: no quadratic regulator off-shell: quadratic regulators exist most general solution known, here 1 Z ∆S = d d x φ p2r(p2) φ − ¯ pr= (p2) − r(p2) F 2 k 2 susy ) cut-off functions for component fields related truncation: leading order local potential approximation, Z d 1 µ i ¯= 1 2 1 00 ¯ 0 Γk = d x 2 @µφ∂ φ + 2 @ − 2 F + 2 Wk (φ) γ∗ − Wk (φ)F Georg Bergner, Jens Braun, Holger Gies, Daniel Litim, Marianne Mastaler, Franziska Synatschke-Czerwonka Phys. Rev. D81 (2010) 125001; D80 (2009) 085007; D80 (2009) 101701 JHEP 0903 (2009) 028; ArXiv 1107.3011; including NLO Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 11 / 31 Flow of super potential project flow onto F =) flow equation for Wk : k d−1 W 00(φ) @ (φ) = − k k Wk 2 00 2 Ad k + Wk (φ) nonlinear PDE 1 02 Wk =) flow for scalar potential Vk = 2 Wk k = Λ: classical potential k ! 0: convex effective potential flow into susy broken phase: Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 12 / 31 10-4x10 k=Λ 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 k=Λ 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 10-4x10 8 ) φ ( 6 2 k W' 4 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 φ Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 13 / 31 Fixed point structure in 2 dimensions dimensionless quantities t = log(k=Λ); Wk (φ) = kwt (φ) 1 w 00(φ) @ (φ) + (φ) = − t t wt wt 00 2 4π 1 + wt (φ) nonlinear PDE 0 even wt ) susy-breaking possible, Taylor series 0 2 2 4 6 wt (φ) = λt (φ − at ) + b4;t φ + b6;t φ + ::: =) 1-system of coupled ODE’s fixed points: @t w∗ = 0 =) 1-system of algebraic equations Andreas Wipf (FSU Jena) Renormalization Group Flows and Supersymmetry 14 / 31 fixed points continued p linearization: a∗=1= 2π always unstable 2n keep terms up to b2n;t φ =) 2n non-Gaussian fixed points ∗ ∗ ∗ ± λp; b4;p;:::; b2n;p ; p = 1;:::; n ∗ ∗ ordering λn > λn−1 > : : : =) ∗ 2 λn : 1 IR-unstable direction at ∗ λn−1 : 2 IR-unstable directions ∗ λn−2 : 3 IR-unstable directions .
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