Introduction to Lattice Supersymmetry

Transcription

1 Introduction to Lattice Supersymmetry Simon Catterall Syracuse University Introduction to Lattice Supersymmetry p. 1

2 Motivation Motivation: SUSY theories - cancellations between fermions/bosons soft U.V behavior. Higgs light m 2 H log(λ) More tractable analytically toy models for understanding confinement and chiral symmetry breaking Key component of string theory remove tachyon of bosonic string. Generalizations of AdSCFT (p + 1) SYM and type II strings with Dp-branes. Usually assume symmetry holds at high energy must break nonperturbatively at low energies lattice. Introduction to Lattice Supersymmetry p. 2

3 Problems Extension of Poincare symmetry: {Q,Q} = γ.p. Broken by lattice. Equivalently: Leibniz rule does not hold for difference operators Fermion doubling - n B n F. Wilson terms break SUSY. Consequence: Naively discretized classical action breaks SUSY. Effective action picks up (many) SUSY violating operators. Generically some of relevant. Couplings must be fine tuned as a 0. Introduction to Lattice Supersymmetry p. 3

4 Solutions Just do it. Certain simple cases eg. N = 1 SYM in 4D single counterterm with Wilson fermions. For D < 4 finite number of divergences occuring at small numbers of loops. For special class of theories can find novel discretizations which preserve one or more SUSY s exactly. Two approaches: Orbifold methods. SYM case. Twisted formulations. Equivalent...? Introduction to Lattice Supersymmetry p. 4

5 Overview Motivation/Problems. Witten s SUSYQM. Naive discretization. Fine tuning. Modification to maintain exact SUSY. Nicolai maps. Topological/twisted field theory interpretation. Generalizations. SYMQM, sigma models. Lifting to 2D. Wess Zumino models. Twisting in 2D. Kähler-Dirac fermions. N = 2 SYM. Lifting to 4D. N = 4 SYM. Orbifold constructions Introduction to Lattice Supersymmetry p. 5

6 Witten s SUSYQM S = dt 1 2 ( dφ dt ) P (φ) ψ dψ i i dt + iψ 1ψ 2 P (φ) Invariant under 2 SUSYs: δ A φ = ψ 1 ǫ A δ B φ = ψ 2 ǫ B δ A ψ 1 = dφ dt ǫ A δ B ψ 1 = ip ǫ B δ A ψ 2 = ip ǫ A δ B ψ 2 = dφ dt ǫ B Homework Problem 1: verify these invariances Introduction to Lattice Supersymmetry p. 6

7 Continuum variation Find: δ A S = dt iǫ ( P dψ 2 dt + dφ ) dt P ψ 2 Need to integrate by parts and use Leibniz to get zero. Problem for lattice. Notice that δa 2 = δ2 B = dt d acting on any field. Example of SUSY algebra since H = dt d. Introduction to Lattice Supersymmetry p. 7

8 Naive discretization Place fields on sites of (periodic) 1D lattice. Replace dt t a and replace dt d by (doubler free) backward difference. a µ f x = f(x) f(x µ) Now find: δ A S L = t iǫ ( P ψ 2 + φp ψ 2 ) where rescaled x by a 1 2. Naively invariant as a 0. But expect radiative corrections... Introduction to Lattice Supersymmetry p. 8

9 Boson/fermion masses - naive m F m B 12.0 M=10.0, G= a Figure 1: P = mφ + gφ 3, m = 10.0, g = Introduction to Lattice Supersymmetry p. 9

10 Radiative corrections Points to note: Any Feynman graph which is convergent in U.V can be discretized naively (Reisz theorem). Restrict attention to superficially divergent continuum graphs. In previous example only one of these. One loop fermion contribution to boson mass. Introduction to Lattice Supersymmetry p. 10

11 Radiative corrections II Continuum: Σ cont = 6g π a π a dp 2π ip + m p 2 + m 2 Actually convergent (p p symmetry) ( ) ( ) 1 π 1 Σ cont = 6g π tan 1 6g 2ma 2 + O(ma) Lattice result is Σ latt = 6g L L 1 k=0 2i sin ( πk L )ei( πk L ) + ma sin 2 ( πk L ) + (ma)2 6g! Introduction to Lattice Supersymmetry p. 11

12 Radiative corrections III If take a 0 after doing sum get twice the result! Homework Problem 2. Convince yourself of this! D = D s m W. Would be doublers have mass O(1/a) and make an additional contribution to integral (don t decouple from small loops) Restore SUSY need to add counterterm S L S L + t 3gφ 2 SUSY broken but regained now as a 0. Introduction to Lattice Supersymmetry p. 12

13 Intuitive argument Consider using lattice derivative: D s + r 2 m W Doubler mass M = m + 2r/a. Consider limit where r << 1. Then ma << Ma << 1. Lattice integral is approx: Σ = = 1 π π a π a dp 2π ( tan 1 m p 2 + m 2 + π a π a π 2ma + tan 1 dp 2π M p 2 + M 2 ) π 2Ma Introduction to Lattice Supersymmetry p. 13

14 Masses - counterterm corrected Figure 2: P = mφ + gφ 3, m = 10.0, g = Introduction to Lattice Supersymmetry p. 14

15 Exact SUSY Can do better. Find combination of SUSY s that can be preserved on lattice. Notice that: δ A S L = iδ B P φ δ B S L = iδ A P φ Thus x x (δ A + iδ B )S L = (δ A + iδ B ) O where O = t P φ So can find δs exact of form S exact L = t 1 2 ( φ) P 2 + P φ + ψ( + P )ψ Introduction to Lattice Supersymmetry p. 15

16 Exact SUSY II Where ψ = 1 2 (ψ 1 + iψ 2 ) ψ = 1 2 (ψ 1 iψ 2 ) and the new supersymmetry acts: δφ = ψǫ δψ = 0 δψ = ( φ + P (φ))ǫ Notice: δ 2 = 0 now. No translations. S b L = x ( φ + P (φ) ) 2 Introduction to Lattice Supersymmetry p. 16

17 Masses - exact SUSY bosons fermions m phys 14.0 M=10.0, G= a Figure 3: Boson and fermion masses vs lattice spacing for supersymmetric action Introduction to Lattice Supersymmetry p. 17

18 Nicolai map Partition function: Z = DφDψDψe S = Dφdet ( + P ) e S B Change variables to N = φ + P (φ) Jacobian is ( det Ni φ j ). Cancels fermionic determinant! Z = dn i e N i 2 Details of P(φ) disappeared! Z is a topological invariant. Simple argument: < S B >= 1 2 N d.o.f i Introduction to Lattice Supersymmetry p. 18

19 Ward identities Classical invariance of action replaced by relationships between correlation functions of form Choosing O = ψ x φ y we find < δo >= 0 < ψ x ψ y > + < ( φ + P ) x φ y >= 0 Expect other SUSY δ = 1 2 (δ A iδ B ) broken. Restored in continuum limit without fine tuning. Introduction to Lattice Supersymmetry p. 19

20 ¼º ¼º ¼º ¼º ¼ ¼º½ ¼º¾ ¼º ¼º ¼º ¼º ¼º ¼º ¼º Ward identities II ½ ¼º¼¼ Ï Ö ¹Á ÒØ ØÝ ½ m = 10 g = 800 Ï Ö ¹Á ÒØ ØÝ ¾ m = 10 g = 800 ¼º¼¼ ¼º¼¼¾ R (1) x R (2) x ¼ ¼º¼¼½ ¹¼º¼¼¾ ¹¼º¼¼ ¹¼º¼¼ ¼ ½¼ ½ ¾¼ x ÔÓ ÒØ Ð ØØ Figure 4: m = 10.0, g = 800.0, from Kaestner et al. Introduction to Lattice Supersymmetry p. 20

21 Topological field theory Actually δ 2 = 0 for the field ψ only by using EOM. Can render symmetry nilpotent off-shell by introducing auxiliary field Qφ = ψ Qψ = 0 Qψ = B QB = 0 Note: absorbed ǫ into variation δ and renamed it Q. Also S b L = x B( φ + P ) 1 2 B2 Introduction to Lattice Supersymmetry p. 21

22 TQFT II Remarkably: S L = Q x ψ( φ P 1 2 B) The action is Q-exact. Like BRST? Consider bosonic model with S(φ) = 0. Invariant under φ φ + ǫ topological symmetry. Quantize: pick gauge function N = 0 and introduce Fadeev-Popov factor Z = Dφdet( N φ )e 1 2α N 2 (φ) Interpret ψ,ψ as ghost fields (α = 1) recover our model! Introduction to Lattice Supersymmetry p. 22

23 Moral of the story Fine tuning problems can be handled in D < 4 by perturbative lattice cals. Even D = 4 N = 1 using chirally improved actions. In some cases can do better find combinations of the supersymmetries in (some) SUSY models which are nilpotent. (Twisted) reformulations are closely connected to construction of TQFT. Actions are Q-exact. Easy to translate to lattice. Simulation can be done with (R)HMC algs. and good agreement with theory Introduction to Lattice Supersymmetry p. 23