PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

where P µ is the four-momentum generator of spacetime translation.
The single particle state of supersymmetric theory fall into irreducible representations
ofthe supersymmetric algebra, called supermultiplets. Each supermultiplet contains
both fermionic and bosonic states, which are commonly known as superpartners of each
other. Each supermultiplet or superfield contains equal number of bosonic and fermionic
degrees of freedom with their masses to be equal. The supermultiplet which contains
chiral fermions and gauge bosons are denoted as chiral superfield and vector superfield,
respectively. In supersymmetry, a chiral superfield contains a Weyl fermion, a scalar and
an auxiliary scalar field component denoted as F, whereas the vector superfield consists of
the vector boson, its fermionic superpartner and an auxiliary scalar field D. Each MSSM
chiral superfield is represented as,
ˆΦ = φ+ √ 2θ˜φ+θθF φ , (1.22)
where φ, ˜φ are the scalar and fermionic fields and Fφ is the auxiliary field. The standard
model is extended to the minimal supersymmetric standard model by introducing
scalar and fermionic superpartners to each of the fermions and scalar of the standard
model, respectively. The matter chiral superfield content of the MSSM and their
SU(3) C
×SU(2) L
×U(1) Y
property are as follows,
) (Ûi
(ũLi ˆQ i = ∼ ˆD i
u Li
˜d Li d Li
)
≡ (3,2, 1 3 ), ˆLi =
( ) )
ˆνi
(˜νLi ν
≡
Li
≡ (1,2,−1),
Ê i ẽ Li e Li
Û c i ≡ (ũ c i u c i ) ≡ (¯3,1,− 4 3 ), ˆDc i ∼ ( ˜d c i d c i ) ≡ (¯3,1, 2 3 ) and Êc i ∼ (ẽc i e c i ) ≡ (1,1,+2).
The Higgs chiral supermultiplets of the MSSM are,
) )
Ĥ u =
(Ĥ+
u
Ĥ 0 u
(
h
+
≡ u
˜h+ u
h 0 u
˜h 0 u
∼ (1,2,+1), Ĥ d =
(
Ĥ0 d
Ĥ − d
The gauge multiplet of the MSSM corresponds to,
) ( ) h
0
∼ d
˜h0 d
h − ˜
d
h − ≡ (1,2,−1).
d
V A
S ∼ (gA ˜g A ) ≡ (8,1,0), V i W ∼ (Wi ˜Wi ) ≡ (1,3,0), V Y ∼ (B ˜B) ≡ (1,1,0).
The Lagrangian of the MSSM is
∫ ∫ ∫
L = d 2 θ W + d 2 θ W +
∫
d 2 θd 2 θ K+
∫
d 2 θ WW +
d 2 θ W W, (1.23)
where K and W are the gauge invariant Kähler potential and superpotential respectively.
The form of the gauge invariant Kähler potential is as the following
K = ˆΦ † e 2gV ˆΦ, (1.24)
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