Introduction toSupersymmetry
Unreasonable effectiveness of the SM
LYukawa = − yt√2H0tLtR + h.c.
H0 = 〈H0〉+ h0 = v + h0
mt = yt v√2
0
R ,L
t t
h h0
Figure 1: The top loop contribution to the Higgs mass term.
−iδm2h|top = (−1)Nc
∫d4k
(2π)4 Tr[−iyt√
2i
6k−mt
(−iy∗t√
2
)i
6k−mt
]= −2Nc|yt|2
∫d4k
(2π)4k2+m2
t
(k2−m2t )2
k0 → ik4, k2 → −k2E
−iδm2h|top = iNc|yt|2
8π2
∫ Λ2
0dk2Ek2
E(k2E−m
2t )
(k2E+m2
t )2
x = k2E +m2
t
δm2h|top = −Nc|yt|2
8π2
∫ Λ2
m2tdx(
1− 3m2t
x + 2m4t
x2
)= −Nc|yt|2
8π2
[Λ2 − 3m2
t ln(
Λ2+m2t
m2t
)+ . . .
]
Lscalar = −λ2 (h0)2(|φL|2 + |φR|2)− h0(µL|φL|2 + µR|φR|2)−m2
L|φL|2 −m2R|φR|2
0
φ , φRL
h h0
Figure 2: Scalar boson contribution to the Higgs mass term via thequartic coupling.
0
Rφ ,
Lφ
h h0
Figure 3: Scalar boson contribution to the Higgs mass term via thetrilinear coupling.
−iδm2h|2 = −iλN
∫d4k
(2π)4
[i
k2−m2L
+ ik2−m2
R
]δm2
h|2 = λN16π2
[2Λ2 −m2
L ln(
Λ2+m2L
m2L
)−m2
R ln(
Λ2+m2R
m2R
)+ . . .
].
−iδm2h|3 = N
∫d4k
(2π)4
[(−iµL i
k2−m2L
)2
+(−iµR i
k2−m2R
)2]
δm2h|3 = − N
16π2
[µ2L ln
(Λ2+m2
L
m2L
)+ µ2
R ln(
Λ2+m2R
m2R
)+ . . .
].
If N = Nc and λ = |yt|2 then Λ2 cancels
If mt = mL = mR and µ2L = µ2
R = 2λm2t log Λ are canceled as well
SUSY will guarantee these relations
Coleman-Mandula
Golfand-Lichtman
Haag-Lopuszanski-Sohnius
SUSY algebra
Qα, Q†α = 2σµααPµ,
σµαα = (1, σi) σµαα = (1,−σi)
σ1 =(
0 11 0
)σ2 =
(0 −ii 0
)σ3 =
(1 00 −1
)[Pµ, Qα] = [Pµ, Q
†α] = 0
[Qα, R] = Qα [Q†α, R] = −Q†α
H = P 0 = 14 (Q1Q
†1 +Q†1Q1 +Q2Q
†2 +Q†2Q2)
(−1)F |boson〉 = +1 |boson〉(−1)F |fermion〉 = −1 |fermion〉
(−1)F , Qα = 0∑i |i〉〈i| = 1
so
∑i〈i|(−1)FP 0|i〉 = 1
4
(∑i〈i|(−1)FQQ†|i〉+
∑i〈i|(−1)FQ†Q|i〉
)= 1
4
(∑i〈i|(−1)FQQ†|i〉+
∑ij〈i|(−1)FQ†|j〉〈j|Q|i〉
)= 1
4
(∑i〈i|(−1)FQQ†|i〉+
∑ij〈j|Q|i〉〈i|(−1)FQ†|j〉
)= 1
4
(∑i〈i|(−1)FQQ†|i〉+
∑j〈j|Q(−1)FQ†|j〉
)= 1
4
(∑i〈i|(−1)FQQ†|i〉 −
∑j〈j|(−1)FQQ†|j〉
)= 0.
SUSY:Qα|0〉 = 0
implies that the vacuum energy vanishes
〈0|H|0〉 = 0
SUSY breaking:Qα|0〉 6= 0
and the vacuum energy is positive
〈0|H|0〉 6= 0
(b)V
φ
VV
φ
φ φsdf
sdfasd
V(a)
(c) (d)
SUSY representationsmassive particle rest frame: pµ = (m,~0).
Qα, Q†α = 2mδααQα, Qβ = 0Q†α, Q
†β = 0
Clifford vacuum:
|Ωs〉 = Q1Q2|m, s′, s′3〉,Q1|Ωs〉 = Q2|Ωs〉 = 0
massive multiplet:
|Ωs〉Q†1|Ωs〉, Q
†2|Ωs〉
Q†1Q†2|Ωs〉
massive “chiral” multiplet:
state s3
|Ω0〉 0Q†1|Ω0〉, Q†2|Ω0〉 ± 1
2
Q†1Q†2|Ω0〉 0
massive vector multiplet:
state s3
|Ω 12〉 ± 1
2
Q†1|Ω 12〉, Q†2|Ω 1
2〉 0, 1, 0,−1
Q†1Q†2|Ω 1
2〉 ± 1
2
Massless particlesframe: pµ = (E, 0, 0,−E)
Q1, Q†1 = 4E
Q2, Q†2 = 0
Qα, Qβ = 0Q†α, Q
†β = 0
Clifford vacuum:
|Ωλ〉 = Q1|E, λ′〉,Q1|Ωλ〉 = 0
〈Ωλ|Q2Q†2|Ωλ〉+ 〈Ωλ|Q†2Q2|Ωλ〉 = 0
〈Ωλ|Q2Q†2|Ωλ〉 = 0
massless supermultipletstate helicity|Ωλ〉 λ
Q†1|Ωλ〉 λ+ 12
CPT invariance requires:
state helicity|Ω−λ− 1
2〉 −λ− 1
2
Q†1|Ω−λ− 12〉 −λ
massless chiral multipletstate helicity|Ω0〉 0
Q†1|Ω0〉 12
include CPT conjugate states:
state helicity|Ω− 1
2〉 − 1
2
Q†1|Ω− 12〉 0
massless vector multipletstate helicity|Ω 1
2〉 1
2
Q†1|Ω 12〉 1
and its CPT conjugate:
state helicity|Ω−1〉 −1
Q†1|Ω−1〉 − 12
Superpartnersfermion ↔ sfermionquark ↔ squarkgauge boson ↔ gauginogluon ↔ gluino
Extended SUSY
Qaα, Q†αb = 2σµααPµδ
ab
Qaα, Qbβ = 0Q†αa, Q
†βb = 0
where
a, b = 1, . . . ,N
U(N )R R-symmetry
massless multiplets: pµ = (E, 0, 0,−E)
Qa1 , Q†1b = 4Eδab ,
Qa2 , Q†2b = 0.
general massless multipletstate helicity degeneracy|Ωλ〉 λ 1
Q†1a|Ωλ〉 λ+ 12 N
Q†1aQ†1b|Ωλ〉 λ+ 1 N (N − 1)/2
......
...Q†11Q
†12 . . . Q
†1N |Ωλ〉 λ+N/2 1
N = 2 massless vector multipletstate helicity degeneracy|Ω−1〉 −1 1
Q†|Ω−1〉 − 12 2
Q†Q†|Ω−1〉 0 1
with the addition of the CPT conjugate:
state helicity degeneracy|Ω0〉 0 1
Q†|Ω0〉 12 2
Q†Q†|Ω0〉 1 1
built from one N = 1 vector multiplet and one N = 1 chiral multiplet.
N = 2 Hypermultipletstate helicity degeneracy|Ω− 1
2〉 − 1
2 1 χαQ†|Ω− 1
2〉 0 2 φ
Q†Q†|Ω− 12〉 1
2 1 ψ†α
gauge-invariant mass term: ψαχα
N = 2 is vector-like
N = 3 massless supermultipletstate helicity degeneracy|Ω−1〉 −1 1
Q†|Ω−1〉 − 12 3
Q†Q†|Ω−1〉 0 3Q†Q†Q†|Ω−1〉 1
2 1
plus CPT conjugate
state helicity degeneracy|Ω− 1
2〉 − 1
2 1Q†|Ω− 1
2〉 0 3
Q†Q†|Ω− 12〉 1
2 3Q†Q†Q†|Ω− 1
2〉 1 1
N = 3 is vector-like
N = 4 massless vector supermultipletstate helicity R|Ω−1〉 −1 1
Q†|Ω−1〉 − 12 4
Q†Q†|Ω−1〉 0 6Q†Q†Q†|Ω−1〉 1
2 4Q†Q†Q†Q†|Ω−1〉 1 1
vector-like theory
Massive SupermultipletsQaα, Q
†αb = 2mδααδ
ab
state spin|Ωs〉 s
Q†αa|Ωs〉 s+ 12
Q†αaQ†βb|Ωs〉 s+ 1
...Q†11Q
†21Q
†12Q
†22 . . . Q
†1NQ
†2N |Ωλ〉 s
N = 2 massive supermultipletstate (dR, 2j + 1)|Ω0〉 (1, 1)
Q†|Ω0〉 (2, 2)Q†Q†|Ω0〉 (3, 1) + (1, 3)
Q†Q†Q†|Ω0〉 (2, 2)Q†Q†Q†Q†|Ω0〉 (1, 1)
16 states: five of spin 0, four of spin 12 , and one of spin 1.
N = 4 massive supermultipletstate (R, 2j + 1)|Ω0〉 (1, 1)
Q†|Ω0〉 (4, 2)Q†Q†|Ω0〉 (10, 1) + (6, 3)
Q†Q†Q†|Ω0〉 (20, 2) + (4, 4)Q†Q†Q†Q†|Ω0〉 (20′, 1) + (15, 3) + (1, 5)
Q†Q†Q†Q†Q†|Ω0〉 (20, 2) + (4, 4)Q†Q†Q†Q†Q†Q†|Ω0〉 (10, 1) + (6, 3)
Q†Q†Q†Q†Q†Q†Q†|Ω0〉 (4, 2)Q†Q†Q†Q†Q†Q†Q†Q†|Ω0〉 (1, 1)
which contains 256 states, including eight spin 32 states and one spin 2
state
Central Charges
Qaα, Q†αb = 2σµααPµδ
ab
Qaα, Qbβ = 2√
2εαβZab
Q†αa, Q†βb = 2
√2εαβZ
∗ab
where
ε = iσ2
for N = 2
Qaα, Q†αb = 2σµααPµδ
ab
Qaα, Qbβ = 2√
2εαβεabZQ†αa, Q
†βb = 2
√2εαβεabZ
Defining
Aα = 12
[Q1α + εαβ
(Q2β
)†]Bα = 1
2
[Q1α − εαβ
(Q2β
)†]reduces the algebra to
Aα, A†β = δαβ(M +√
2Z)Bα, B†β = δαβ(M −
√2Z)
〈M,Z|BαB†α|M,Z〉+ 〈M,Z|B†αBα|M,Z〉 = (M −√
2Z) ,
M ≥√
2Z
for M =√
2Z (short multiplets): Bα produces states of zero norm
M >√
2Z (long multiplets)
short (BPS) multiplet:
state 2j + 1|Ω0〉 1
A†|Ω0〉 2(A†)2|Ω0〉 1
state 2j + 1|Ω 1
2〉 2
A†|Ω 12〉 1 + 3
(A†)2|Ω 12〉 2
short multiplet has 8 states as opposed to 32 states for the correspondinglong multiplet
BPS state:
M =√
2Z
is exact