Introduction to Supersymmetry Unreasonable effectiveness of the SM - PowerPoint PPT Presentation

Introduction to Supersymmetry

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

� � � � � � d 4 k − iy ∗ − iy t − iδm 2 i i � h | top = ( − 1) N c (2 π ) 4 Tr √ √ t k − m t k − m t 2 2 k 2 + m 2 d 4 k − 2 N c | y t | 2 � = t (2 π ) 4 ( k 2 − m 2 t ) 2 k 0 → ik 4 , k 2 → − k 2 E � Λ 2 iN c | y t | 2 k 2 E ( k 2 E − m 2 t ) − iδm 2 dk 2 h | top = E ( k 2 E + m 2 8 π 2 0 t ) 2 x = k 2 E + m 2 t � Λ 2 − N c | y t | 2 � 1 − 3 m 2 + 2 m 4 � δm 2 h | top = t dx t t m 2 8 π 2 x 2 x � Λ 2 − 3 m 2 � Λ 2 + m 2 � � − N c | y t | 2 = t ln + . . . t 8 π 2 m 2 t

2 ( h 0 ) 2 ( | φ L | 2 + | φ R | 2 ) − h 0 ( µ L | φ L | 2 + µ R | φ R | 2 ) − λ L scalar = L | φ L | 2 − m 2 − m 2 R | φ R | 2 φ , φ R L 0 0 h h Figure 2: Scalar boson contribution to the Higgs mass term via the quartic coupling.

φ , φ L R 0 0 h h Figure 3: Scalar boson contribution to the Higgs mass term via the trilinear coupling. � � d 4 k − iδm 2 � i i h | 2 = − iλN L + (2 π ) 4 k 2 − m 2 k 2 − m 2 R � � Λ 2 + m 2 � � Λ 2 + m 2 � � 2Λ 2 − m 2 δm 2 λN − m 2 h | 2 = L ln R ln + . . . . L R 16 π 2 m 2 m 2 L R �� � 2 � � 2 � d 4 k − iδm 2 i i � h | 3 = N − iµ L − iµ R + (2 π ) 4 k 2 − m 2 k 2 − m 2 L R � � Λ 2 + m 2 � � Λ 2 + m 2 � � δm 2 N µ 2 + µ 2 h | 3 = − L ln R ln + . . . . L R m 2 m 2 16 π 2 L R

If N = N c and λ = | y t | 2 then Λ 2 cancels If m t = m L = m R and µ 2 L = µ 2 R = 2 λm 2 t log Λ are canceled as well SUSY will guarantee these relations

Coleman-Mandula

Golfand-Lichtman

Haag-Lopuszanski-Sohnius

SUSY algebra { Q α , Q † α } = 2 σ µ α P µ , ˙ α ˙ αα = (1 , − σ i ) σ µ (1 , σ i ) σ µ ˙ = α ˙ α � 0 � 0 � 1 � � � 1 − i 0 σ 1 = σ 2 = σ 3 = − 1 1 0 i 0 0 [ P µ , Q α ] = [ P µ , Q † α ] = 0 ˙ [ Q † α , R ] = − Q † [ Q α , R ] = Q α ˙ α ˙ H = P 0 = 1 4 ( Q 1 Q † 1 + Q † 1 Q 1 + Q 2 Q † 2 + Q † 2 Q 2 )

( − 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 i � i | ( − 1) F QQ † | i � + � i � i | ( − 1) F Q † Q | i � �� � � = 4 �� � 1 i � i | ( − 1) F QQ † | i � + � ij � i | ( − 1) F Q † | j �� j | Q | i � = 4 �� � 1 i � i | ( − 1) F QQ † | i � + � ij � j | Q | i �� i | ( − 1) F Q † | j � = 4 �� � 1 i � i | ( − 1) F QQ † | i � + � j � j | Q ( − 1) F Q † | j � = 4 �� � 1 i � i | ( − 1) F QQ † | i � − � j � j | ( − 1) F QQ † | j � = 4 = 0 .

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

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

SUSY representations massive particle rest frame: p µ = ( m,� 0). { Q α , Q † α } = 2 m δ α ˙ α ˙ { Q α , Q β } = 0 { Q † α , Q † β } = 0 ˙ ˙ Clifford vacuum: | Ω s � = Q 1 Q 2 | m, s ′ , s ′ 3 � , Q 1 | Ω s � = Q 2 | Ω s � = 0 massive multiplet: | Ω s � Q † 1 | Ω s � , Q † 2 | Ω s � Q † 1 Q † 2 | Ω s �

massive “chiral” multiplet: state s 3 | Ω 0 � 0 Q † 1 | Ω 0 � , Q † ± 1 2 | Ω 0 � 2 Q † 1 Q † 2 | Ω 0 � 0 massive vector multiplet: state s 3 ± 1 | Ω 1 2 � 2 Q † 2 � , Q † 1 | Ω 1 2 | Ω 1 2 � 0 , 1 , 0 , − 1 Q † 1 Q † ± 1 2 | Ω 1 2 � 2

Massless particles frame: p µ = ( E, 0 , 0 , − E ) { Q 1 , Q † 1 } = 4 E { Q 2 , Q † 2 } = 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 2 Q † 2 | Ω λ � + � Ω λ | Q † 2 Q 2 | Ω λ � = 0 � Ω λ | Q 2 Q † 2 | Ω λ � = 0

massless supermultiplet state helicity | Ω λ � λ Q † λ + 1 1 | Ω λ � 2 CPT invariance requires: state helicity − λ − 1 | Ω − λ − 1 2 � 2 Q † 1 | Ω − λ − 1 2 � − λ

massless chiral multiplet state helicity | Ω 0 � 0 Q † 1 1 | Ω 0 � 2 include CPT conjugate states: state helicity − 1 | Ω − 1 2 � 2 Q † 1 | Ω − 1 2 � 0

massless vector multiplet state helicity 1 | Ω 1 2 � 2 Q † 1 | Ω 1 2 � 1 and its CPT conjugate: state helicity | Ω − 1 � − 1 Q † − 1 1 | Ω − 1 � 2

Superpartners fermion ↔ sfermion ↔ quark squark ↔ gauge boson gaugino gluon ↔ gluino

Extended SUSY α , Q † 2 σ µ { Q a α P µ δ a αb } = ˙ α ˙ 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 ) 1 , Q † { Q a 4 Eδ a 1 b } = b , 2 , Q † { Q a 2 b } = 0 .

general massless multiplet state helicity degeneracy | Ω λ � λ 1 Q † λ + 1 1 a | Ω λ � N 2 Q † 1 a Q † 1 b | Ω λ � λ + 1 N ( N − 1) / 2 . . . . . . . . . Q † 11 Q † 12 . . . Q † 1 N | Ω λ � λ + N / 2 1

N = 2 massless vector multiplet state helicity degeneracy | Ω − 1 � − 1 1 Q † | Ω − 1 � − 1 2 2 Q † Q † | Ω − 1 � 0 1 with the addition of the CPT conjugate: state helicity degeneracy | Ω 0 � 0 1 Q † | Ω 0 � 1 2 2 Q † Q † | Ω 0 � 1 1 built from one N = 1 vector multiplet and one N = 1 chiral multiplet.

N = 2 Hypermultiplet state helicity degeneracy − 1 | Ω − 1 2 � 1 χ α 2 Q † | Ω − 1 2 � 0 2 φ Q † Q † | Ω − 1 1 ψ † ˙ α 2 � 1 2 gauge-invariant mass term: ψ α χ α N = 2 is vector-like

N = 3 massless supermultiplet state helicity degeneracy | Ω − 1 � − 1 1 Q † | Ω − 1 � − 1 3 2 Q † Q † | Ω − 1 � 0 3 Q † Q † Q † | Ω − 1 � 1 1 2 plus CPT conjugate state helicity degeneracy − 1 | Ω − 1 2 � 1 2 Q † | Ω − 1 2 � 0 3 1 Q † Q † | Ω − 1 2 � 3 2 Q † Q † Q † | Ω − 1 2 � 1 1 N = 3 is vector-like

N = 4 massless vector supermultiplet state helicity R | Ω − 1 � − 1 1 Q † | Ω − 1 � − 1 4 2 Q † Q † | Ω − 1 � 0 6 Q † Q † Q † | Ω − 1 � 1 4 2 Q † Q † Q † Q † | Ω − 1 � 1 1 vector-like theory

Massive Supermultiplets α , Q † { Q a α δ a αb } = 2 m δ α ˙ ˙ b state spin | Ω s � s Q † s + 1 αa | Ω s � ˙ 2 Q † αa Q † βb | Ω s � s + 1 ˙ ˙ . . . Q † 11 Q † 21 Q † 12 Q † 22 . . . Q † 1 N Q † 2 N | Ω λ � s

N = 2 massive supermultiplet state ( d R , 2 j + 1) | Ω 0 � (1 , 1) Q † | Ω 0 � (2 , 2) Q † Q † | Ω 0 � (3 , 1) + (1 , 3) Q † Q † Q † | Ω 0 � (2 , 2) Q † Q † Q † Q † | Ω 0 � (1 , 1) 16 states: five of spin 0, four of spin 1 2 , and one of spin 1.

N = 4 massive supermultiplet state ( R , 2 j + 1) | Ω 0 � ( 1 , 1) Q † | Ω 0 � ( 4 , 2) Q † Q † | Ω 0 � ( 10 , 1) + ( 6 , 3) Q † Q † Q † | Ω 0 � ( 20 , 2) + ( 4 , 4) Q † Q † Q † Q † | Ω 0 � ( 20 ′ , 1) + ( 15 , 3) + ( 1 , 5) Q † Q † Q † Q † Q † | Ω 0 � ( 20 , 2) + ( 4 , 4) Q † Q † Q † Q † Q † Q † | Ω 0 � ( 10 , 1) + ( 6 , 3) Q † Q † Q † Q † Q † Q † Q † | Ω 0 � ( 4 , 2) Q † Q † Q † Q † Q † Q † Q † Q † | Ω 0 � ( 1 , 1) which contains 256 states, including eight spin 3 2 states and one spin 2 state

Central Charges α , Q † 2 σ µ { Q a α P µ δ a αb } = √ α ˙ ˙ b { Q a α , Q b 2 ǫ αβ Z ab β } = 2 √ { Q † αa , Q † β Z ∗ βb } = 2 2 ǫ ˙ α ˙ ˙ ˙ ab where ǫ = iσ 2 for N = 2 α , Q † 2 σ µ { Q a α P µ δ a αb } = √ ˙ α ˙ b { Q a α , Q b 2 ǫ αβ ǫ ab Z β } = 2 √ { Q † αa , Q † βb } = 2 2 ǫ ˙ β ǫ ab Z α ˙ ˙ ˙

Defining � � † � � 1 Q 1 Q 2 A α = α + ǫ αβ β 2 � � † � � 1 Q 1 Q 2 α − ǫ αβ B α = β 2 reduces the algebra to √ { A α , A † β } = δ αβ ( M + 2 Z ) √ { B α , B † β } δ αβ ( M − = 2 Z ) √ � M, Z | B α B † α | M, Z � + � M, Z | B † α B α | M, Z � = ( M − 2 Z ) , √ M ≥ 2 Z √ for M = 2 Z (short multiplets): B α produces states of zero norm √ M > 2 Z (long multiplets)

short (BPS) multiplet: state 2 j + 1 | Ω 0 � 1 A † | Ω 0 � 2 ( A † ) 2 | Ω 0 � 1 state 2 j + 1 | Ω 1 2 � 2 A † | Ω 1 2 � 1 + 3 ( A † ) 2 | Ω 1 2 � 2 short multiplet has 8 states as opposed to 32 states for the corresponding long multiplet BPS state: √ M = 2 Z is exact