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

Introduction to Supersymmetry

Unreasonable effectiveness of the SM

LYukawa = − yt

√ 2H0tLtR + h.c.

H0 = H0 + h0 = v + h0 mt = yt v

√ 2

R

,

L

t t h h

Figure 1: The top loop contribution to the Higgs mass term.

−iδm2

h|top

= (−1)Nc

• d4k

(2π)4 Tr

• −iyt

√ 2 i

• k−mt
• −iy∗

t

√ 2

• i
• k−mt
• =

−2Nc|yt|2

d4k (2π)4 k2+m2

t

(k2−m2

t )2

k0 → ik4, k2 → −k2

E

−iδm2

h|top

=

iNc|yt|2 8π2

Λ2 dk2

E k2

E(k2 E−m2 t )

(k2

E+m2 t )2

x = k2

E + m2 t

δm2

h|top

= − Nc|yt|2

8π2

Λ2

m2

t dx

• 1 − 3m2

t

x

+ 2m4

t

x2

• =

− Nc|yt|2

8π2

• Λ2 − 3 m2

t ln

• Λ2+m2

t

m2

t

• + . . .

Lscalar = − λ

2 (h0)2(|φL|2 + |φR|2) − h0(µL|φL|2 + µR|φR|2)

−m2

L|φL|2 − m2 R|φR|2

φ , φR

L

h h

Figure 2: Scalar boson contribution to the Higgs mass term via the quartic coupling.

R

φ ,

L

φ h h

Figure 3: Scalar boson contribution to the Higgs mass term via the trilinear coupling. −iδm2

h|2 = −iλN

• d4k

(2π)4

• i

k2−m2

L +

i k2−m2

R

• δm2

h|2 = λN 16π2

• 2Λ2 − m2

L ln

• Λ2+m2

L

m2

L

• − m2

R ln

• Λ2+m2

R

m2

R

• + . . .
• .

−iδm2

h|3 = N

• d4k

(2π)4

• −iµL

i k2−m2

L

2 +

• −iµR

i k2−m2

R

2 δm2

h|3 = − N 16π2

• µ2

L ln

• Λ2+m2

L

m2

L

• + µ2

R ln

• Λ2+m2

R

m2

R

• + . . .
• .

If N = Nc and λ = |yt|2 then Λ2 cancels If mt = mL = mR and µ2

L = µ2 R = 2λm2 t log Λ are canceled as well

SUSY will guarantee these relations

Coleman-Mandula

Golfand-Lichtman

Haag-Lopuszanski-Sohnius

SUSY algebra

{Qα, Q†

˙ α} = 2σµ α ˙ αPµ,

σµ

α ˙ α

= (1, σi) σµ ˙

αα = (1, −σi)

σ1 = 1 1

• σ2 =

−i i

• σ3 =

1 −1

• [Pµ, Qα] = [Pµ, Q†

˙ α] = 0

[Qα, R] = Qα [Q†

˙ α, R] = −Q† ˙ α

H = P 0 = 1

4(Q1Q† 1 + Q† 1Q1 + Q2Q† 2 + Q† 2Q2)

(−1)F |boson = +1 |boson (−1)F |fermion = −1 |fermion {(−1)F , Qα} = 0

• i |ii| = 1

so

• ii|(−1)FP 0|i

=

1 4

• ii|(−1)FQQ†|i +

ii|(−1)FQ†Q|i

• =

1 4

• ii|(−1)FQQ†|i +

iji|(−1)FQ†|jj|Q|i

• =

1 4

• ii|(−1)FQQ†|i +

ijj|Q|ii|(−1)FQ†|j

• =

1 4

• ii|(−1)FQQ†|i +

jj|Q(−1)FQ†|j

• =

1 4

• ii|(−1)FQQ†|i −

jj|(−1)FQQ†|j

• =

0.

SUSY:

Qα|0 = 0 implies that the vacuum energy vanishes 0|H|0 = 0

SUSY breaking:

Qα|0 = 0 and the vacuum energy is positive 0|H|0 = 0

(b) V φ V V φ φ φ

sdf sdfasd

V (a) (c) (d)

SUSY representations

massive particle rest frame: pµ = (m, 0). {Qα, Q†

˙ α}

= 2 m δα ˙

α

{Qα, Qβ} = {Q†

˙ α, Q† ˙ β}

= 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 Q†

1|Ω0, Q† 2|Ω0

± 1

2

Q†

1Q† 2|Ω0

massive vector multiplet: state s3 |Ω 1

2

± 1

2

Q†

1|Ω 1

2 , Q†

2|Ω 1

2

0, 1, 0, −1 Q†

1Q† 2|Ω 1

2

± 1

2

Massless particles

frame: pµ = (E, 0, 0, −E) {Q1, Q†

1}

= 4E {Q2, Q†

2}

= {Qα, Qβ} = {Q†

˙ α, Q† ˙ β}

= Clifford vacuum: |Ωλ = Q1|E, λ′, Q1|Ωλ = 0 Ωλ|Q2Q†

2|Ωλ + Ωλ|Q† 2Q2|Ωλ = 0

Ωλ|Q2Q†

2|Ωλ = 0

massless supermultiplet

state helicity |Ωλ λ Q†

1|Ωλ

λ + 1

2

CPT invariance requires: state helicity |Ω−λ− 1

2

−λ − 1

2

Q†

1|Ω−λ− 1

2

−λ

massless chiral multiplet

state helicity |Ω0 Q†

1|Ω0 1 2

include CPT conjugate states: state helicity |Ω− 1

2

− 1

2

Q†

1|Ω− 1

2

massless vector multiplet

state helicity |Ω 1

2

1 2

Q†

1|Ω 1

2

1 and its CPT conjugate: state helicity |Ω−1 −1 Q†

1|Ω−1

− 1

2

Superpartners

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

Extended SUSY

{Qa

α, Q† ˙ αb}

= 2σµ

α ˙ αPµδa b

{Qa

α, Qb β}

= {Q†

˙ αa, Q† ˙ βb}

= where a, b = 1, . . . , N U(N)R R-symmetry massless multiplets: pµ = (E, 0, 0, −E) {Qa

1, Q† 1b}

= 4Eδa

b ,

{Qa

2, Q† 2b}

= 0.

general massless multiplet

state helicity degeneracy |Ωλ λ 1 Q†

1a|Ωλ

λ + 1

2

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 multiplet

state helicity degeneracy |Ω−1 −1 1 Q†|Ω−1 − 1

2

2 Q†Q†|Ω−1 1 with the addition of the CPT conjugate: state helicity degeneracy |Ω0 1 Q†|Ω0

1 2

2 Q†Q†|Ω0 1 1 built from one N = 1 vector multiplet and one N = 1 chiral multiplet.

N = 2 Hypermultiplet

state helicity degeneracy |Ω− 1

2

− 1

2

1 χα Q†|Ω− 1

2

2 φ Q†Q†|Ω− 1

2

1 2

1 ψ† ˙

α

gauge-invariant mass term: ψαχα N = 2 is vector-like

N = 3 massless supermultiplet

state helicity degeneracy |Ω−1 −1 1 Q†|Ω−1 − 1

2

3 Q†Q†|Ω−1 3 Q†Q†Q†|Ω−1

1 2

1 plus CPT conjugate state helicity degeneracy |Ω− 1

2

− 1

2

1 Q†|Ω− 1

2

3 Q†Q†|Ω− 1

2

1 2

3 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

2

4 Q†Q†|Ω−1 6 Q†Q†Q†|Ω−1

1 2

4 Q†Q†Q†Q†|Ω−1 1 1 vector-like theory

Massive Supermultiplets

{Qa

α, Q† ˙ αb}

= 2 m δα ˙

αδa b

state spin |Ωs s Q†

˙ αa|Ωs

s + 1

2

Q†

˙ αaQ† ˙ βb|Ωs

s + 1 . . . Q†

11Q† 21Q† 12Q† 22 . . . Q† 1N Q† 2N |Ωλ

s

N = 2 massive supermultiplet

state (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 1

2, and one of spin 1.

N = 4 massive supermultiplet

state (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 3

2 states and one spin 2

state

Central Charges

{Qa

α, Q† ˙ αb}

= 2σµ

α ˙ αPµδa b

{Qa

α, Qb β}

= 2 √ 2ǫαβZab {Q†

˙ αa, Q† ˙ βb}

= 2 √ 2ǫ ˙

α ˙ βZ∗ ab

where ǫ = iσ2 for N = 2 {Qa

α, Q† ˙ αb}

= 2σµ

α ˙ αPµδa b

{Qa

α, Qb β}

= 2 √ 2ǫαβǫabZ {Q†

˙ αa, Q† ˙ βb}

= 2 √ 2ǫ ˙

α ˙ βǫabZ

Defining Aα =

1 2

• Q1

α + ǫαβ

• Q2

β

† Bα =

1 2

• Q1

α − ǫαβ

• Q2

β

† 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 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 = √ 2Z is exact