466
E. Rabinovici
standard model particles carrying Baryonic and Leptonic numbers. The symmetry
also in its continuous U(1) version turns out to play a possible role in the possibilities
to spontaneously break SUSY. It is called R symmetry.
R-symmetry is a global U(1) symmetry that does not commute with the supersymmetry. Its discrete version is called R parity. The action of the R-symmetry on a
superfield with R-character n as follows.
RΦ (θ, x) = exp (2inα) Φ (exp (−iαθ) , x)
(8.29)
RΦ
θ, x
= exp (−2inα) Φ
exp
iαθ
, x
(8.30)
Since the R-charge does not commute with the supersymmetry, the component
fields of the chiral field have different R-charges. For a superfield with R-character
n, the R-charges of the component fields may be read off as follows:
R (lowest component of Φ) = R(A) ≡ n, R (ψ) = n − 1, R(F ) = n − 2
(8.31)
The R-charge of the Grassmann variables is given by:
R (θ α ) = 1, R (dθ α ) = −1
(8.32)
with, barred variables having opposite R charge. The kinetic term ΦΦ is an R
invariant. (θ θθθ is an invariant.) For the potential term,
d
2 θW
(8.33)
to have zero R charge requires that R(W) = 2. For the resulting mass term from
W =
1
2 mΦ 2 ,
mψψ + m
2
|A|
2 ,
(8.34)
to have vanishing R-charge requires
R (Φ) = R(A) = 1, R (ψ) = 0
(8.35)
Adding the cubic term:
W 3 =
λ
3
Φ
3
(8.36)