Introduction to Supersymmetry

Transcription

1 Introduction to Supersymmetry

2 Unreasonable effectiveness of the SM L Yukawa = y t H 0 t L t R + h.c. H 0 = H 0 + h 0 = v + h 0 m t = y t v t, L t R h 0 h 0 Figure 1: The top loop contribution to the Higgs mass term.

3 iδm h [ top = ( 1)N d 4 k c (π) Tr iyt 4 = N c y t d 4 k (π) 4 k +m t (k m t ) ( ) i iy t k m t ] i k m t k 0 ik 4, k k E iδm h top = in c y t 8π Λ 0 dk E k E (k E m t ) (k E +m t ) x = ke + m t δm h top = N c y t Λ ( 8π dx 1 3m m t t x = N c y t 8π [Λ 3 m t ln ) + m4 t ( x ) Λ +m t m t ] +...

4 L scalar = λ (h0 ) ( φ L + φ R ) h 0 (µ L φ L + µ R φ R ) m L φ L m R φ R φ L, φ R h 0 h 0 Figure : Scalar boson contribution to the Higgs mass term via the quartic coupling.

5 φ, L φ R h 0 h 0 Figure 3: Scalar boson contribution to the Higgs mass term via the trilinear coupling. iδm h = iλn [ ( δm h = λn 16π Λ m L ln Λ +m L m L iδm h 3 = N d 4 k (π) 4 [ ( iµ L [ ( ) δm h 3 = N 16π µ L ln Λ +m L m L [ d 4 k i (π) 4 k m L ) ] i + k m R m R ln ( Λ +m R m R ) ( i k m + iµ R L + µ R ln ( Λ +m R m R ) ] ) ] i k m R ) ] +....

6 If N = N c and λ = y t then Λ cancels If m t = m L = m R and µ L = µ R = λm t log Λ are canceled as well SUSY will guarantee these relations

7 Coleman-Mandula

8 Golfand-Lichtman

9 Haag-Lopuszanski-Sohnius

10 SUSY algebra σ 1 = ( {Q α, Q α } = σµ α α P µ, σ µ α α = (1, σ i ) σ µ αα = (1, σ i ) ) σ = ( 0 i i 0 ) [P µ, Q α ] = [P µ, Q α ] = 0 σ 3 = [Q α, R] = Q α [Q α, R] = Q α ( ) H = P 0 = 1 4 (Q 1Q 1 + Q 1 Q 1 + Q Q + Q Q )

11 ( 1) F boson = +1 boson ( 1) F fermion = 1 fermion {( 1) F, Q α } = 0 i i i = 1 so i i ( 1)F P 0 i = 1 4 ( i i ( 1)F QQ i + i i ( 1)F Q Q i ) ( = 1 4 i i ( 1)F QQ i + ) ij i ( 1)F Q j j Q i ( = 1 4 i i ( 1)F QQ i + ) ij j Q i i ( 1)F Q j ( = 1 4 i i ( 1)F QQ i + ) j j Q( 1)F Q j ( = 1 4 i i ( 1)F QQ i ) j j ( 1)F QQ j = 0.

12 SUSY: Q α 0 = 0 implies that the vacuum energy vanishes 0 H 0 = 0 SUSY breaking: Q α 0 0 and the vacuum energy is positive 0 H 0 0

13 (a) V sdfasd (b) V φ φ (c) V (d) V φ sdf φ

14 SUSY representations massive particle rest frame: p µ = (m, 0). {Q α, Q α } = m δ α α {Q α, Q β } = 0 {Q α, Q β} = 0 Clifford vacuum: Ω s = Q 1 Q m, s, s 3, Q 1 Ω s = Q Ω s = 0 massive multiplet: Ω s Q 1 Ω s, Q Ω s Q 1 Q Ω s

15 massive chiral multiplet: state s 3 Ω 0 0 Q 1 Ω 0, Q Ω 0 ± 1 Q 1 Q Ω 0 0 massive vector multiplet: state s 3 Ω 1 ± 1 Q 1 Ω 1, Q Ω 1 0, 1, 0, 1 Q 1 Q Ω 1 ± 1

16 frame: p µ = (E, 0, 0, E) Massless particles {Q 1, Q 1 } = 4E {Q, Q } = 0 {Q α, Q β } = 0 {Q α, Q β} = 0 Clifford vacuum: Ω λ = Q 1 E, λ, Q 1 Ω λ = 0 Ω λ Q Q Ω λ + Ω λ Q Q Ω λ = 0 Ω λ Q Q Ω λ = 0

17 massless supermultiplet CPT invariance requires: state helicity Ω λ λ Q 1 Ω λ λ + 1 state helicity Ω λ 1 λ 1 Q 1 Ω λ 1 λ

18 massless chiral multiplet include CPT conjugate states: state helicity Ω 0 0 Q 1 Ω 1 0 state helicity Ω 1 1 Q 1 Ω 1 0

19 and its CPT conjugate: massless vector multiplet state helicity Ω 1 1 Q 1 Ω 1 1 state helicity Ω 1 1 Q 1 Ω 1 1

20 Superpartners fermion sfermion quark squark gauge boson gaugino gluon gluino

21 Extended SUSY {Q a α, Q αb } = σµ α α P µδ a b {Q a α, Q b β } = 0 {Q αa, Q βb } = 0 where a, b = 1,..., N U(N ) R R-symmetry massless multiplets: p µ = (E, 0, 0, E) {Q a 1, Q 1b } = 4Eδa b, {Q a, Q b } = 0.

22 general massless multiplet state helicity degeneracy Ω λ λ 1 Q 1a Ω λ λ + 1 N Q 1a Q 1b Ω λ λ + 1 N (N 1)/... Q 11 Q 1... Q 1N Ω λ λ + N / 1

23 N = massless vector multiplet state helicity degeneracy Ω Q Ω 1 1 Q Q Ω with the addition of the CPT conjugate: state helicity degeneracy Ω Q 1 Ω 0 Q Q Ω built from one N = 1 vector multiplet and one N = 1 chiral multiplet.

24 N = Hypermultiplet state helicity degeneracy Ω χ α Q Ω 1 0 φ Q Q Ω ψ α gauge-invariant mass term: ψ α χ α N = is vector-like

25 N = 3 massless supermultiplet plus CPT conjugate state helicity degeneracy Ω Q Ω Q Q Ω Q Q Q 1 Ω 1 1 state helicity degeneracy Ω Q Ω Q Q Ω Q Q Q Ω N = 3 is vector-like

26 N = 4 massless vector supermultiplet state helicity R Ω Q Ω Q Q Ω Q Q Q 1 Ω 1 4 Q Q Q Q Ω vector-like theory

27 Massive Supermultiplets {Q a α, Q αb } = m δ α αδ a b state spin Ω s s Q αa Ω s s + 1 Q αaq βb Ω s s + 1. Q 11 Q 1 Q 1 Q... Q 1N Q N Ω λ s

28 N = massive supermultiplet state (d R, j + 1) Ω 0 (1, 1) Q Ω 0 (, ) Q Q Ω 0 (3, 1) + (1, 3) Q Q Q Ω 0 (, ) Q Q Q Q Ω 0 (1, 1) 16 states: five of spin 0, four of spin 1, and one of spin 1.

29 N = 4 massive supermultiplet state (R, j + 1) Ω 0 (1, 1) Q Ω 0 (4, ) Q Q Ω 0 (10, 1) + (6, 3) Q Q Q Ω 0 (0, ) + (4, 4) Q Q Q Q Ω 0 (0, 1) + (15, 3) + (1, 5) Q Q Q Q Q Ω 0 (0, ) + (4, 4) Q Q Q Q Q Q Ω 0 (10, 1) + (6, 3) Q Q Q Q Q Q Q Ω 0 (4, ) Q Q Q Q Q Q Q Q Ω 0 (1, 1) which contains 56 states, including eight spin 3 state states and one spin

30 Central Charges {Q a α, Q αb } = σµ α α P µδ a b {Q a α, Q b β } = ɛ αβ Z ab {Q αa, Q βb } = ɛ α βz ab where ɛ = iσ for N = {Q a α, Q αb } = σµ α α P µδ a b {Q a α, Q b β } = ɛ αβ ɛ ab Z {Q αa, Q βb } = ɛ α βɛ ab Z

31 Defining A α = 1 B α = 1 [Q 1α + ɛ αβ ( Q β ) ] ( ) ] [Q 1α ɛ αβ Q β reduces the algebra to {A α, A β } = δ αβ(m + Z) {B α, B β } = δ αβ(m Z) M, Z B α B α M, Z + M, Z B αb α M, Z = (M Z), M Z for M = Z (short multiplets): B α produces states of zero norm M > Z (long multiplets)

32 short (BPS) multiplet: state j + 1 Ω 0 1 A Ω 0 (A ) Ω 0 1 state j + 1 Ω 1 A Ω (A ) Ω 1 short multiplet has 8 states as opposed to 3 states for the corresponding long multiplet BPS state: M = Z is exact