ON ULTRAVIOLET STRUCTURE OF 6D SUPERSYMMETRIC GAUGE THEORIES Ft. Lauderdale, December 18, 2015 PLAN Philosophical introduction Technical interlude Something completely different (if I have time) 0-0
PHILOSOPHICAL INTRODUCTION: Is supergravity finite? Einstein s gravity and its SUSY extentions are nonrenormalizable. like Fermi s theory! Cross sections grow with energy, σ G 2 E 2. Unitary limit: impossibility of perturbative calculations beyond E m p = G 1/2. Power divergences in the loops, M gg gg = M tree gg gg(1 + c G 0 Λ 2 + ). G 1 eff hence = G 1 0 (1 + c G 0Λ 2 + ). Power divergences appear with a physical regularization, like lattice regularization. No continuum limit of the lattice path integral. 0-1
People (Z. Bern et al) calculate it with dimensional regularization. Only logarithmic divergences are left. ln Λ may multiply higher-dimensional structures in the effective Lagrangian (higher-dimensional counterterms). Many of the latter vanish on mass shell and are irrelevant. 0-2
example: Einstein s gravity. L tree G 1 0 R. Canonical dimension d = 2. Eq. mot. R µν = 0. d = 4: R 2, R 2 µν Fantasme: L 1 loop = G 1 0 R(1 + c G 0Λ 2 ) + c R 2 ln Λ but: R 2 and R 2 µν vanish on mass shell. Do not contribute in the amplitudes. Ergo Einstein s gravity is finite in one loop, if disregarding power divergences. Logarithms appear in 2 loops, L 2 loop = G 1 0 R [ 1 + c G 0 Λ 2 + c (G 0 Λ 2 ) 2] +c G 0 ln ΛR µναβ R µνγδ R αβ γδ N = 8 4D SUGRA is finite through 6 loops if disregarding power divergences (no appropriate counterterms). Directly checked by Bern s group through 4 loops. 0-3
STRUCTURE OF COUNTERTERMS IN SUSY THEORIES Example: N = 1 4D SYM. Component action, d = 4 S = d x 1 2 F µνf µν i λ σ µ µ λ Invariant under nonlinear SUSY transformations, δa µ = i λ σ µ ξ + i ξ σ µ λ δλ = σ µν ξf µν Superfield formulation includes auxiliary fields D. Extra term D 2 in L. In d = 6 structures like W α D2 W α d 2 θ + c.c., auxiliary fields become dynamical. Extra term ( µ D) 2. 0-4
Impossible to write down a higher-dimensional off-shell supersymmetric action that involves only A µ and λ α. Similar, but more complicated in N = 2 4D SYM theories. N = 4 SYM The component action is invariant under nonlinear SUSY transformations. No superfield formulation no off-shell invariant action with d > 4 N = 8 SUGRA Tree d = 2 Lagrangian is invariant under nonlinear SUSY transformations. 2. No off-shell invariant counterterms with d > But: Starting from the 7th loop, there are counterterms which enjoy extended supersymmetry on mass shell : The variation is proportional to the equations of motion of L d=2. These on-shell, but not off-shell invariant structures contribute to the amplitudes! 0-5
Counterterms in N = (1,1) 6D SYM. based on [G. Bossard + E. Ivanov + A.S., arxiv:1509.08027, to be published in JHEP] Fields: (i) bosons: A M, real φ A i (i, A = 1, 2); (ii) fermions: λ a, ψ a left and right (a = 1, 2, 3, 4); (iii) auxiliary fields D ik, F A i. Component Lagrangian [f] = m 1, L = 1 2f 2 F 2 MN + - color trace. We studied counterterms in the language of N = (1, 0) off-shell and N = (1, 1) on-shell harmonic superspaces. 0-6
CONSCLUSIONS No off-shell N = (1, 1) invariants with d = 6, 8 etc. 6. No on-shell SUSY invariant of dimension d = No on-shell N = (1, 1) d = 8 invariant respecting N = (1, 0) off-shell invariance. Relaxing the latter requirement, an on-shell N = (1, 1) d = 8 invariant can be written. It does not contribute to the amplitudes, but is relevant to some fancy effective string/brane actions. At the level d = 10 (3 loops), there exist two different on-shell invariants. Experimental fact: only one of them is relevant for the amplitudes. 0-7
N = (1, 0) harmonic superspace z = (x M, θ ak ) (x M, θ ±a ) (a = 1, 2, 3, 4; k = 1, 2) with θ ±a = u ± k θak (note pseudoreality: C a b (θb i ) = ǫ ik θ ak ) Spinor SUSY derivatives D + a = θ a +iθ+b ab, D a = θ +a iθ b ab. (V ab = 1 2 (γm ) ab V M.) {D + a, D b } = 2i ab Harmonic derivatives D ++ = u +i u i, D = u i u + i, 0-8
Gauge superfield V ++ Grassmann analyticity D + a V ++ = 0. Introduce V satisfying the zero harmonic curvature condition D ++ V D V ++ + [V ++, V ] = 0 Introduce W +a = 1 6 ǫabcd D + b D+ c D + d V F ++ = 1 24 ǫabcd D +a D + b D+ c D + d V Dimension: [V ++ ] = [V ] = 1, [W +a ] = 3/2, [F ++ ] = m 2. 0-9
N = (1, 0) ACTIONS d = 4 B. Zupnik, 1986 S = 1 f 2 n=2 ( 1) n n d 6 x d 8 θ du 1...du n {V ++ (z, u 1 )...V ++ (z, u n )} (u + 1 u+ 2 )...(u+ n u + 1 ), (1) d = 6 E. Ivanov + B. Zupnik + A.S., 2005 S (6) SY M = 1 2g 2 d 6 x d 4 θ + du (F ++ ) 2. Dimensionless constant. Chiral anomaly nonrenormalizable A.S., 2006 Vanishes on the mass shell of (1). 0-10
S = d = 8 d 6 xd 4 θ + du ε abcd W +a W +b W +c W +d. Generically, D + a W +a = 4F ++ 0 not an off-shell invariant. Is invariant on mass shell (F ++ = 0). d = 10 Two off-shell invariants, S (10) 1 = ε abcd and S (10) 2 = ǫ abcd d 6 xd 8 θdu W +a W b W +c W d d 6 xd 8 θdu W +a W b W +c W d 0-11
MATTER MULTIPLETS q +A, A = 1, 2, D + a q +A = 0, Involves 4 real scalars and right-handed spinors. d = 4 action S = 1 2f 2 d 6 x d 4 θ + du q +A (D ++ + V ++ )q + A. Modified equations of motion F ++ + 1 2 [q+a, q + A ] = 0 d = 6 action S = 1 2g 2 d 6 x d 4 θ + du (F ++ + 12 ) 2 [q+a, q +A ] Gives a renormalizable higher-derivative 6D theory. Vanishes on the mass shell of the d = 4 action and is not a relevant counterterm. E. Ivanov + A.S., 2006; A.S. 2006 0-12
On-shell harmonic N = (1, 1) superfields. (Non-harmonic: Howe+ Sierra+ Townsend, 1983. In the superspace {x M θ ak, ˆθ A a } ) Introduce besides u ± k the harmonics uˆ+ A, uˆ A. Superspace: {x M θ +a, θ a ; θ ˆ+ a, θ ˆ a } Introduce the spinor covariant derivatives + a = D + a + A + a, ˆ+a = D ˆ+a + B aˆ+ Their algebra ( central basis) { + a, + b } = { ˆ+a, ˆ+b } = 0, { + a, ˆ+b } = δaφ b +ˆ+, [D ++, + a ] = [D ˆ+ˆ+, + a ] = 0 [D ++, aˆ+ ] = [D ˆ+ˆ+, aˆ+ ] = 0. with φ +ˆ+ satisfying + a φ +ˆ+ = aˆ+ φ +ˆ+ = D ++ φ +ˆ+ = D ˆ+ˆ+ φ +ˆ+ = 0, 0-13
Abelian case φ +ˆ+ = ϕ +ˆ+ θ +a ψ ˆ+ a θ ˆ+ a λ +a + i 6 θ ˆ+ a θ +b F a b iθ +a θ +b ab ϕ ˆ+ iθ ˆ+ a θ ˆ+ b ab ϕ +ˆ + iθ ˆ+ a θ +b θ +c bc λ a +iθ +a θ ˆ+ b θ ˆ+ c bc ψ ˆ a θ ˆ+ a θ ˆ+ b θ+c θ +d ab cd ϕ ˆ, with F a b = (σmn ) a b F MN and λ, ψ, φ satisfying ab λ a = ab ψ a = 0, ϕ = 0 This is an on-shell superfield (analogy with q + satisfying D + q + = D ++ q + = 0). Expression of φ +ˆ+ via the (1, 0) superfields in the hat-analytic basis (B aˆ+ = 0). φ +ˆ+ = q +ˆ+ θ ˆ+ a W +a iθ ˆ+ a θ ˆ+ b ab q +ˆ + 1 6 εabcd θ ˆ+ a θ ˆ+ b θ ˆ+ c [D + q ˆ d, q +ˆ ] 1 24 εabcd θ ˆ+ a θ ˆ+ b θ ˆ+ c θ ˆ+ d, [q +ˆ, q ˆ ]]. (2) where q +ˆ+ = q +A uˆ+ A, q+ˆ = q +A uˆ A. 0-14
S 8 = inv. action, d = 8 d 6 x d 4 θ + d 4 θ ˆ+ dudû (φ +ˆ+ ) 4 inv. actions, d = 10 S 10 1 = d 6 x d 8 θd 4 θ ˆ+ dudû (φ +ˆ+ ) 2 (φ ˆ+ ) 2 and S 10 1 = d 6 x d 8 θd 4 θ ˆ+ dudû (φ +ˆ+ ) 2 (φ ˆ+ ) 2 only the first structure is seen in the perturbative calculations. Explicit expressions in terms of N = (1, 0) off-shell superfields were derived. 0-15
Effective Lagrangians and supersymmetry Question: Where extended SUSY is gone? Answer: It is still there, but only for the whole sum, S eff = 1 f 2 (1+c f2 Λ 2 + )S (4) +c f 4 ln Λ S (10) + and not for the individual counterterms. 0-16
Example: Maximal SQM L = 1 2 A A I A A I g2 4 fabe f CDE A A I A B J A C I A D J ig 2 fabc λ A α(γ I ) αβ λ B β A C I, I = 1,...,9, λ A α are real fermions, α = 1,...,16. 16 real supercharges. Consider SU(2), f ABC ǫ ABC. Abelian valley, A A I = A Ic A. Born-Oppenheimer expansion in γ BO = 1 g A 3. 0-17
Leading order: Equations of motion: Invariant under L 2 = 1 2 A 2 I + i 2 λ α λ α. (3) Ä I = 0, λ α = 0. δ 0 A I = iǫγ I λ, δ 0 λ = A I Γ I ǫ, No potential is generated. No corrections to the metric. 0-18
Higher-derivative corrections L 4 (Ȧ2 I )2 A 7 + fermion terms, (the indices 2,4,... count the number of derivatives). suppressed as γ 3 BO compared to (3) L 4 is not invariant under δ 0. The full is invariant under L eff = L 2 + L 4 + L 6 + δ = δ 0 + δ 2 + δ 4 +. 0-19
The full expression for L 4 and δ 2 was derived in Y. Kazama + T. Muramatsu, 2001. Complicated. L 6 and δ 4 not known. From δl = 0, we derive δ 2 L 2 + δ 0 L 4 = 0, δ 4 L 2 + δ 2 L 4 + δ 0 L 6 = 0,... Hence δ 0 L 4 eq. mot. Vanishes on mass shell. But δ 0 L 6 does not! The terms eq. mot. in L 4 can be killed by a field redefinition. Then L 4 greatly simplifies and acquires the form (H. Nicolai + J. Plefka, 2000) (v I = A I ) mass shell L4 (v 2 I )2 A I i v J v 2 λγ IJ λ 7, 0-20