How to write SUSY Lagrangians
1st step
Take your favorite Lagrangian written in terms of fields
2nd step
Replace
Field (ϕ ,ψ , Aµ ) ⇒ Superfield (Φ, V )
3rd step
Replace
Action = ∫ d x L( x)
4
Grassmannian integration in superspace
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∫d
4
x d θ L ( x, θ , θ )
4
∫ dθα = 0, ∫ θ β dθα = δαβ
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Superfields
SUSY Lagrangian (Matter)
L = ∫ d 2θ d 2θ Φ i+ Φ i + ∫ d 2θ (λi Φ i + 12 mij Φ i Φ j + 13 yijk Φ i Φ j Φ k ) + h.c.]
Kinetic term
Components
Superpotential
µ
L = i∂ µ ψ i σ ψ i + Ai* , Ai + Fi * Fi
no derivatives
+ [λi Fi + mij ( Ai Fj − 12 ψ iψ j ) + yijk ( Ai Aj Fk − ψ iψ j Ak ) + h.c.]
Constraint
δL
= Fk* + λk + mik Ai + yijk Ai Aj = 0
δ Fk
Fk
µ
L = i∂ µ ψ i σ ψ i + Ai* , Ai − 12 mijψ iψ j − 12 mij*ψ iψ j
*
− yijkψ iψ j Ak − yijk
ψ iψ j Ak* − V ( Ai , Aj )
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Pre-conference school, SUSY'07
V = Fk* Fk
2
SUSY Lagrangians (gauge)
Gauge fields
L=
1
4
α
2
µν
µ
1
1
d
θ
W
W
+
d
θ
W
W
=
D
−
F
F
−
i
λσ
Dµ λ
α
α
2
4 µν
∫
∫
2
α
2
Gauge transformation
Φ→e
− ig Λ
+
+ ig Λ +
Φ, Φ → Φ e
, V → V + i (Λ − Λ + )
Gauge invariant interaction with matter (covariant derivative)
Φ + Φ → Φ + e gV Φ
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Gauge Invariant SUSY Lagrangian
α
∫ d θ Tr(W Wα ) + ∫ d θ Tr(W W α )
+ ∫ d θ d θ Φ (e ) Φ + ∫ d θ W (Φ ) + ∫ d θ W (Φ )
LSUSY YM =
2
2
ia
LSUSY YM = − Fµν F
1
4
α
2
1
4
a
a µν
gV a
b
1
4
b
i
2
2
2
i
i
a
µ
− iλ σ Dµ λ + 12 D a D a
a
µ
+ (∂ µ Ai − igvµ T Ai ) (∂ µ Ai − igvµ T Ai ) − iψ i σ (∂ µψ i − igvµa T aψ i )
a
a
a
†
a
a
− D gA T Ai − i 2 gA T λ ψ i + i 2 gψ i T λ Ai + Fi † Fi
a
†
i
a
†
i
a
a
a
2
∂W
∂W † 1 ∂ 2W
∂
W
1
ψ iψ j − 2 † † ψ iψ
Fi + † Fi − 2
+
∂Ai
∂Ai
∂Ai ∂A j
∂Ai ∂A j
∂W
D = − gA T Ai , Fi = −
→ V= 12 D a D a + Fi † Fi
∂Ai
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a
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j
†
i
a
Spontaneous Breaking of SUSY
E =< 0 | H | 0 >
Energy
E=
1
4
∑
α
=1,2
j
< 0 |{Qα , Qα }| 0 >=
i
E =< 0 | H | 0 >≠ 0
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j
β
{Qα , Q } = 2δ (σ )αβ Pµ
i
1
4
ij
µ
2
|
Q
|
0
>
|
≥0
∑ α
α
if and only if
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Qα | 0 >≠ 0
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Mechanism of SUSY Breaking
Fayet-Iliopoulos (D-term) mechanism
(in Abelian theory)
O’Raifertaigh (F-term) mechanism
∆L = ξV |θθ θ θ = ξ ∫ d 4θ V = ξ D ≠ 0
W (Φ ) = λΦ 3 + mΦ1Φ 2 + g Φ 3Φ12
F1* = mA2 + 2 gA1 A2
F2* = mA1
F3* = λ + gA12
< Fi >≠ 0
∑
bosons
D-term
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mi2 =
∑
mi2
fermions
F-term
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Minimal Supersymmetric
Standard Model (MSSM)
SUSY: # of fermions = # of bosons
N=1 SUSY:
(ϕ ,ψ ) (λ , Aµ )
SM: 28 bosonic d.o.f. & 90 (96) fermionic d.o.f.
There are no particles in the SM that can be superpartners
SUSY associates known bosons with new fermions
and known fermions with new bosons
Even number of the Higgs doublets – min = 2
Cancellation of axial anomalies (in each generation)
64
Tr Y 3 = 3( 271 + 271 − 27
+ 278 ) − 1 − 1 + 8 = 0
colour u L d L u R d R ν L eL eR
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Higgsinos
-1+1=0
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Particle Content of the MSSM
Superfield
Gauge
Bosons
Fermions
g a g a
gluino
gluino
k
k k (w
±
± ± , z
,
wino
zino
w
Weak W (W , Z ) wino, zino w ( w , z ) )
Hypercharge B(γ ) binobino
b (γ )b (γ )
Ga
Vk
V′
gluon g a
Matter
LLi == ((νν,,ee))L
Li = (ν , e) L
Li
i
L
sleptons
leptons
EEii = eRR
Ei = eR
Ei
i == ((uu,,dd)) L
Q
Q
Qi = (u , d ) L
Qi
i
L
quarks U i = u Rc
U i squarks UUi i ==uuRR
c
==dd
Di
DD
D
=
d
ii
RR
i
R
Higgs
H1
H2
Higgses
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{H
H1
2
higgsinos
{
H1 1
H
H 2
H
2
Pre-conference school,
SUSY'07
SU c (3) SU L (2) U Y (1)
8
1
0
1
3
0
1
1
0
1
2
−1
1
1
2
3
2
1/ 3
3*
1
−4 / 3
3*
1
2/3
1
1
2
2
−1
1
8
The MSSM Lagrangian
L = Lgauge + LYukawa + LSoftBreaking
The Yukawa Superpotential
Superfields
WR = yU QL H 2U R + yD QL H1 DR + yL LL H1 ER + µ H1 H 2
Yukawa couplings
Higgs mixing term
WNR = λL LL LL ER + λ LLQL DR + µ LL H 2 + λBU R DR DR
'
L
Violate:
Lepton number
'
Baryon number
λL , λL' < 10−6 , λB < 10−9
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Pre-conference school, SUSY'07
These terms are
forbidden in
the SM
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R-parity
R = (−)3( B − L ) + 2 S
B - Baryon Number
L - Lepton Number
S - Spin
The Usual Particle : R = + 1
SUSY Particle :
R= -1
The consequences:
• The superpartners are created in pairs
• The lightest superparticle is stable
e
ip
+
p
e
• The lightest superparticle (LSP)
should be neutral - the best candidatej
is neutralino (photino or higgsino) χ 0
• It can survive from the Big Bang and
form the Dark matter in the Universe
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−
ip
p
χj0
χj0
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Interactions in the MSSM
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Creation of Superpartners at
colliders
Annihilation channel
Gluon fusion, ee, qq scattering
and qg scattering channels
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Decay of Superpartners
squarks
0
i
q L ,R → q + χ i
±
i
q L → q '+ χ i
q L , R → q + gi
sleptons
l → l + χ
i0
i
l L → ν + χ
i±
i
l
chargino χi i± → e + ν e + χi i0
±
0
i
i
χ i → q + q '+ χ i
gluino
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gi → q + q + γ
gi → g + γ
neutralino
Final sates
0
0
i
i
χ i → χ1 + l+ + l−
0
0
i
i
χi → χ1 + q + q '
l +l − + ET
2 jets + E T
±
0
i
i
χ i → χ1 + l± +ν l
γ + ET
0
0
i
i
χ i → χ1 +ν l +ν
ET
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l
13
Soft SUSY Breaking
Hidden
sector
SUSY
MSSM
Messengers
Gravitons, gauge, gauginos, etc
Breaking via F and D terms in a hidden sector
− LSoft = ∑ M i λii λii + ∑ m02i | Ai |2 + + ∑ A ijk Ai A j Ak + ∑ Bij Ai A j
α
i
gauginos
ijk
ij
scalar fields
Over 100 of free parameters !
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We like elegant solutions
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