Supersymmetric Flows for Supersymmetric Field Theories
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1 Supersymmetric Flows for Supersymmetric Field Theories A. Wipf Theoretisch-Physikalisches Institut, FSU Jena in collaboration with G. Bergner, H. Gies, J. Braun, F. Synatschke-Czerwonka Phys. Rev. D81 (2010) ; Phys. Rev. D80 (2009) ; Phys. Rev. D80 (2009) ; JHEP 0903 (2009) 028 ERG 2010: 5th International Conference on the Exact Renormalization Group, September, 2010 Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 1 / 19
2 1 Why apply ERGE to SUSY-theories 2 Supersymmetric Yukawa (Wess-Zumino) models 3 Flow of super-potential 4 2 space-time dimensions 5 3 space-time dimensions Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 2 / 19
3 Supersymmetry (susy) and ERGE particle physics beyond Standard Model = supersymmetry bosons fermions susy-breaking collective condensation phenomena = non-perturbative methods lattice simulations: susy (partially) broken by spacetime lattice, dynamical fermions,... = need complement to lattice studies exact renormalization group? Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 3 / 19
4 Challenges manifest supersymmetric renormalization flow e.g. m boson = m fermion dynamical susy breaking = phase transitions fixed-point structure, temperature effects, equation of state non-renormalization theorems relevant dof at low energies? (cp. Veneziano-Yankielowicz) related (mostly structural) investigations by: Sonoda; Bonini & Vian; Falkenberg & Geyer; Arnone & Yoshida; Arnone& Guerrieri & Yoshida; Rosten, Sonoda & Ulker; Horiskoshi & Aoki & Taniguchi & Terao talk of Synatschke-Czerwonka Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 4 / 19
5 Supersymmetric Yukawa (Wess-Zumino) models minimal supersymmetry real fields (φ, ψ, F ): Lagrangian L = 1 2 µφ µ φ + i ψ 2 / ψ 1 2 F W (φ) ψγ ψ W (φ)f action invariant under susy transformation: δφ = εγ ψ, δψ = (F + iγ / φ)ε, δf = i ε/ ψ eliminate auxiliary field F = L = 1 2 ( φ)2 + i ψ 2 / ψ W 2 (φ) W (φ) ψγ ψ classical Yukawa-model determined by superpotential W W (φ) φ m : m even: susy always unbroken m odd: susy breaking possible Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 5 / 19
6 need supersymmetric regulator for flow equation (Wetterich 93) k Γ k = 1 { [ ] } 1 2 STr Γ (2) k + R k k R k superspace formulation = general cutoff R k depends on two functions r(p 2 ) and s(p 2 ) choose s(p 2 ) = 0 = S k = 1 d d x ( φ p 2 r(p 2 ) φ 2 ψ /pr(p 2 ) ψ r(p 2 ) F 2) susy relates 3 cut-off functions here: 2 and 3 dimensions model with extended supersymmetry talk of Synatschke-Czerwonka Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 6 / 19
7 Flow of super potential this talk: mainly local potential approximation Γ k = d d x ( 1 2 µφ µ φ + i ψ 2 / ψ 1 2 F W k (φ) ψγ ψ W k(φ)f ) project flow to F = flow equation for W k (φ) : NLO: Synatschke, Gies, Wipf k W k (φ) = k d 1 W k (φ) A d k 2 + W k, A (φ)2 2 = 4π, A 3 = 8π 2 Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 7 / 19
8 Fixed point structure in 2 dimensions dimensionless quantities k t, W k (φ) = kw t (φ) allow for susy-breaking: even w t t w t (φ) + w t (φ) = 1 w t (φ) 4π 1 + w t (φ) 2 w t (φ) = λ t (φ 2 a 2 t ) + b 4,t φ 4 + b 6,t φ = system of coupled ODE s a t does not enter equations for higher order couplings λ t, b 2i,t fixed point analysis: (a ) 2 = 1/2π, at 2 : always IR-unstable Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 8 / 19
9 2d-fixed points continued keep terms up to b 2n,t φ 2n = 2n non-gaussian fixed points ordering λ n > λ n 1 >... = λ n : λ n 1 λ n 2 ± ( λ p, b 4,p,..., b 2n,p), p = 1,..., n 1 IR-unstable direction at 2 : 2 IR-unstable directions : 3 IR-unstable directions... root belonging to IR-stable fixed point λ n n λ crit = Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 9 / 19
10 λ R(θ I ) of non-gaussian fixed points, truncation at 2n=16 ± ± ± ± ± ± ± ± odd solutions of nonlinear ODE for u(φ) = w (φ) : (1 u 4 )u = 2u 2 (3 u 2 )u (1 u 2 ) 3 4πu periodic solutions for u (0) 2λ crit, λ crit = previous polynomials converge to periodic solution u crit (0) = 2λ crit: IR-stable fixed point, finite for φ Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 10 / 19
11 next-to leading-order flows wave function renormalization = η = t log Z 2 k θ 0 critical exponent of relevant direction a 2 t (related to W ) new superscaling relation (exact in NLO) ν w = 1 θ 0 = d η 2 superscaling relation at maximally IR-stable fixed point (d=2) 2n η /ν W number of IR-unstable direction = number of nodes of u plus 1 Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 11 / 19
12 Supersymmetry breaking supersymmetric phase: min φ V k=0 (φ) = min φ W k=0 2 (φ) = 0 susy broken: W k=0 (φ) has no node t= W Λ (φ)=λ Λ (φ 2 -a- 2 Λ ) t=-0.6 t= λ Λ =0.1Λ, a- 2 Λ =0.3 t= unbroken SuSy W k 2 (φ)/λ (λ Λ a - 2 Λ )/Λ broken SuSy φ λ Λ /Λ left: flow of a potential V = W 2 with susy breaking, W Λ (φ) = λ Λ (φ 2 ā 2 Λ ) right: phase diagram for couplings specified at Λ, different truncations. φ 10 φ 8 φ 4 φ 2 Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 12 / 19
13 masses of bosons and fermions supersymmetric phase: Zk 4m2 k,boson = W k 2 (χ min /Z k ) = Zk 4m2 k,fermion broken phase:(superscaling) Z 4 k m2 k,boson = W k 1+η/2 (0)W k (0) k t= -9 t=-10 t=-11 5 t=-12 t=-13 broken SuSy unbroken SuSy 4 m k /Λ 3 m/25λ (λ Λ a 2 Λ )/Λ Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 13 / 19
14 Wess-Zumino model in 3 dimensions one Wilson-Fisher fixed point for Yukawa-model LPA, polynomial expansion with J. Braun and F. Synatschke-Czerwonka Wilson-Fisher fixed point from polynomial expansion 2n ±λ ±b4 ±b6 ±b8 ±b10 ±b rapid convergence (contrary to 2 dimensions) a 2 t defines the only IR-unstable direction Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 14 / 19
15 2n critical exponents for different truncations phase diagram from parameter study of W k unbroken SuSy 0.04 (λa 2 ) Λ 0.03 broken SuSy φ 2 φ 4 φ λ Λ Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 15 / 19
16 Finite temperature dp0 summation over Matsubara frequencies sums can be calculated explicitly = two flow equations k W k bos k W k ferm k 2 = = k 2 8π 2 W k k 2 W k 2 (k 2 + W k 2 F bos(t, k) )2 8π 2 W k k 2 W 2 k (k 2 + W 2 k )2 F ferm(t, k) susy breaking by thermal fluctuations (bosons fermions) T = 0: susy broken Z 2 unbroken = study Z 2 breaking at finite T Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 16 / 19
17 Phase diagram unbroken Z 2 T/Λ broken Z λ Λ a 2 Λ finite-temperature phase diagram for fixed λ Λ = 0.8 Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 17 / 19
18 Phase diagram, continued T/Λ broken Z λλ 1.8 λ Λ a 2 Λ finite-temperature phase diagram Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 18 / 19
19 Future supersymmetric O(n) and CP(n) models lattice O(3) CP(1) flow equation for large n supersymmetric gauge theories first studies lattice flow equation R. Flore, D. Körner, C. Wozar M. Masthaler, F. Synatschke-Czerwonka B. Wellegehausen F. Synatschke-Czerwonka Andreas Wipf (FSU Jena) Supersymmetric Flows for Supersymmetric Field Theories 19 / 19
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