From Susy-QM to Susy-LFT Andreas Wipf, TPI Jena Hamburg, 24. April, 2005 in collaboration with: A. Kirchberg, D. Länge, T. Kästner Outline: Generalities From Susy-QM to Susy-LT vacuum sector susy vs. lattice derivatives 1
Motivation, Problems Why Susy? Only possible extension of Poincaré invariance to larger spacetime symmetry (HSL) Physics beyond Standard Model (hierarchy, gauge coupling unification, dark matter candidates) Superstring theory (susy needed for consistent QG?) Tool to obtain results in strongly coupled QFT. Why Susy-QM? Most simple susy-ft IR-dynamics of susy-ft in finite V Matrix theory description of M theory Index theorems, isopectral deformations, integrable systems, susy inspired approximations... 2
Why Susy-LT? Nonperturbative dynamics (spectrum, CSB) Confirm/extend existing results (N = 2 SYM) Check conjectured results (N = 1 SYM) How to deal with fermions Gauge Theories Main focus on N = 1 SYM: L = 1 4 F µνf µν i ψ 2 /Dψ U A (1) anomaly condensate 2Nc 2 N c ground states λλ = cλ 3 e 2πik/N c, c =? (SU(2) and SU(3) : DESY-Münster) hadron spectrum (VY action from WI) mass splitting for m g 0 (Evans et.al) N c αλ soft breaking m f0 m χ m η m g 3
beyond VY (Farrar et.al, Louis et.al) glue/gluinoball mixing (Peetz et.al: small) which type of fermions? Wilson: non-chiral, undoubled, ultra-local, cheap, hermitean susy broken by lattice Wilson term susy and chiral limit at m g 0 condensate, smallest masses, Ward identities: Montvay et.al, Peetz et.al,... Ginsparg-Wilson (overlap, domain walls): chiral, undoubled, local, expensive, non-hermitean no fine-tuning for chiral limit condensate: Kogut et.al N = 2 or N = 4 SYM: 2 or 6 scalars fine tuning gets worse euclidean S B unbounded from below? new approaches (Kaplan et.al, Sugino, Itoh et.al) 4
Problems specific to Lattice regularization No discrete version of susy plethora of unwanted relevant operators, fine tuning models with subset of exact susy? (WZ; SYM with N > 1) No Leibniz rule on lattice Λ: f : Λ, (f, g) = x f(x)g(x) (Df)(x) = y D xy f(y) linear D(fg) = (Df)g f(dg) = D = 0 ultralocal forward/backward derivatives (a=1): ( f µ f)(x) = f(x ˆµ) f(x) ( b µf)(x) = f(x) f(x ˆµ) (f, f µ g) = ( b µf, g). 5
weighted sum: r [ 1, 1] µ r = 1 2 (1 r) µ f 1 2 (1 r) µ b = µ a µ, s antisymmetric and symmetric parts µ a = 1 2 ( µ f µ) b, µ s = r 2 ( µ f µ) b r = 0: antisymmetric and doubling; s µ = r 2 f µ b µ }{{} = µ s µ = r 2 a Wilson fermions S F = ( ψ, D w ψ), D w = iγ µ a µ m }{{} doubling r 2 a }{{} W term Hamiltonian form: continuous t, discrete space h r F = iα i M a i β ( m r 2 a ) = h r F 6
Slac fermions (Weinstein et.al): chiral, no doubling h F = iαm i i slac βm = h F, slac i antisymmetric slac r = 1 r c r = 0 λ k (h F ) m = 0 d = 11 k µ slac non-local, identical spectrum as µ cont Ginsparg-Wilson fermions (e.g. Neuberger) γ 5 D Dγ 5 = adγ 5 D, D = γ 5 Dγ 5 7
Models, Methods and selected results Divergences depend on fermion type: (d, N ) = (2, 2) and (4, 1) Wess-Zumino: tadpoles linearly divergent for W, finite for WG (Fujikawa) No Leibniz rule: - superfield superfield superfield - would be central charges not central: (divw (φ)) (x) x x,i W i (φ) φ(x) φ(x) x i - restoration of rule for a 0 (Fujikawa et.al) Exact (partial) susy on lattice: decrease number of fine tunings applied to d = 2, 4 WZ and extended SYM covariant quantization: typically non-local fermions; 8
sometimes non-local interaction (no Leibniz rule!) - keep subset of extended susy: 4 1 (Catterall) - keep Leibniz rule via noncommutativity (D Adda) (Kaplan, Sugino, Campostrini,... ) Hamiltonian formalism: time continuous, keep spectral subalgebra (?) of {Q I α, Q J β} = 2 ( δ IJ /P αβ iδ αβ Z IJ A iγ αβ ZS IJ ) Z : central charges, Q I α real supercharges Example: WZ in 2 dimensions: Z IJ A = 0, N = 1 : Z S = dx x W N = 2 : F (φ 1 iφ 2 ) = W iu ( ) Z IJ S = σ3 dx x W σ 1 dx x U, 9
Susy-QM Susy FT on finite space lattice = susy-qm lattice derivative: most simple case ( ) 0 A Q =, A = W, A = W, A 0 ( AA ) H = Q 2 0 = 0 A A discretize H: (AA ) L = 2 W 2 W (A A) L = 2 W 2 W discretize Q: isospectral operators A L A L = W 2 W W A L A L = W 2 W W Difference: no Leibniz,, central charge 10
1.4 1.2 1.0 0.8 0.6 E n /E exact n Q 2 V ± = λ 2 x 4 naiv ± 2λx H SLAC Q 2 SLAC 20 40 60 80 120 n H naiv Figure 1: EV of (Q L ) 2 and H L for SLAC derivative and forward-derivative for N = 180, L = 30, λ = 1. non-local SLAC-derivative: - much more accurate for WZ-models (first studies) - further improvement for gauge theories (Slavnov) generalization lattice FT (here N = 2): {ψ, ψ } =, nilpotent complex charge: Q = ψ ( W ), ψ = ( ) 0 1 0 0 2H = {Q, Q } 11
2N dimensions: x 2N, ψ 1,..., ψ 2N Q = W n (x) = n χ(x), {ψ n, ψ m } = δ mn 2N n=1 ψ n ( n n χ) = e χ Q 0 e χ, Q 2 = 0 Hamiltonian of 2N-dimensional N = 2 SQM: H = 1 2 1 2 ( χ, χ) 1 2 χ ψn χ,nm ψ m H = h } {{ h }, h = L 2 ( N times 2 ) 4 SQM Lattice-WZ 2,2 via identifications φ(n) = ( x 2n 1 ), ψ(n) = x 2n ( ψ 2n 1 ) ψ 2n H = (ψ, h F ψ)... = χ = 1 2 (φ, h F φ)... χ real: γ 0 = σ 3, γ 1 = iσ 1, γ = σ 2 12
( ) m x h F = m x = iγ a x mγ 0 iγ 1 s x unusual Wilson term s (N = 1: usual one) Interacting fields h 2 F = ( m 2 ) 2 χ = 1 2 (φ, h F φ) f ( φ(n) ), f = 0. Super-Hamiltonian: H = P 0 Z P 0 = 1 2 (π, π) 1 2 (φ, φ) 1 2 ( φf, φ f) (ψ, h 0 F ψ) ( ψ, γ 0 Γ(φ)ψ ) Γ(φ) = f,11 (φ) iγ f,12 (φ), Yukawa f harmonic f = g φ 1 φ 2 and f = g φ 2 φ 1 13
Would be central charge Z = ( φ g, a xφ) ( φ g, σ 3 s xφ) weak coupling: free massive model for f = 1 2 m(φ2 2 φ 2 1): exist one unique susy ground state (for all ) strong coupling: (Jaffe et.al) Neglect in controlled way H = n h n h n single site operators on L 2 ( 2 4 ) H = 1 2 1 2 ( f, f) ψ f ψ. f(φ) = λ p R (φ 1 iφ 2 ) p Elitzur, Schwimmer: p 1 susy ground states Lattice WZ has (p 1) N susy ground states 14
Can prove: Holds true for all λ 0; e.g. f(φ) = λ p R (φ 1 iφ 2 ) p o(φ p ) = (p 1) N Highly degenerate vacuum sector!! From strong to weak coupling: Step 1: Perturbation Q(ɛ) = Q 0 ɛq 1 hermitean Q i, commute with Γ = Γ, Γ 2 = Q(ɛ)ψ(ɛ) = λ(ɛ)ψ(ɛ) Q 0 ψ 0 = 0, Γψ 0 = ψ 0 λ(0) = 0; formal power series ψ(ɛ) = ψ 0 ɛ k ψ k, λ(ɛ) = k=1 ɛ k λ k k=1 Proposition: (Jaffe, Lesniewski and Lewenstein) as formal power series λ(ɛ) = 0 and Γψ(ɛ) = ψ(ɛ) 15
Step 2: In Majorana representation {Q 1 1, Q 1 1} = {Q 2 1, Q 2 1} = 2(P 0 Z) {Q 1 1, Q 2 2} = 0, Z = ZS 11 = ZS 22 Q 1 1 = (π, ψ 1 ) ( W, ψ 2 ) }{{} B 0 strong coupling B 0 essentially s.a. on D(B 0 ) = C c ( use B 0 to define energy norm ( φ 1, ψ 1 2) ( φ 2, ψ 2 2) }{{} B 1 perturbation 2N ) bounds on a f 2 B 0 f 2, f D(B 0 ) = D( B 0 ) weighted Sobolov space For all λ, ɛ > 0 exists C ɛ > 0 such that D λb 1 f ɛ B 0 f C ɛ f, f D( B 0 ) Kato-Rellich: Q 1 1(λ) s.a. on D( B 0 ) λb 1 is B 0 bounded with arbitrary small bound 16
Q 1 1(ɛ) analytic family (Kato) Spectrum of Q 1 1(ɛ) discrete λ(ɛ) analytic KLW, Annals of Physics 316 (2004) 357 applies to N = 1 in d = 2: deg W = p even: susy unbroken in sc and pt deg W = p odd: susy broken in sc; may be unbroken in pt; example W (φ) = g 2 φ 3 g 0 φ strong coupling: susy broken g 0 (correct) weak coupling: susy for g 0 < 0 applies to N = 2 in d = 2: puzzle (p 1) N susy ground states on finite lattice (p 1) susy ground states in continuum (Jaffe et.al) conjecture H = P 0 Z, Z a 0 c < 0 17
rigoros result independent on choice of N = 1 WZ in 4 dimensions analog results for ground states of Q 2 1 = H P 3. Not applicable to potentials with flat directions!! For arbitrary : would be central charges ZS IJ = σ3 IJ (W, a φ) σ1 IJ (U, a φ) ( γ 0 (U, s φ) ZL IJ = σ3 IJ γ 0 (W, s φ) σ1 IJ i (π, I s φ) i 2 (ψ, I s ψ) ) cont. algebra, no doubling, chiral symmetry Slac derivative favoured (gauge theories?) checked index theorem on lattice of D slac [U] how to implement (detailed balance, ergodic,... ) 18