Perturbative and non-perturbative infrared behavior of ...

3. The Wess-Zumino model
dim U(1)R U(1)Φ dim U(1)R U(1)Φ
Φ 1 1 1 ¯ Φ 1 −1 −1
U −4 4 0 d 4 θ 2 0 0
Dα 1/2 −1 0 ¯ D ˙α 1/2 1 0
D 2 1 −2 0 ¯ D 2 1 2 0
g 0 −1 −3 ¯g 0 1 3
m 1 0 −2 ¯m 1 0 2
κ1 0 0 0 κ2 0 0 0
Table 3.1: Global U(1) charge assignment in superspace
• unless not explicitly shown, the ∗-product does not get deformed at the quantum level;
indeed, the effective action can be written by keeping the product implicit.
3.2 An all loop argument
The renormalizability of the WZ model was further discussed in subsequent papers and an all
loop argument was given [9, 10, 11]. We review here the superspace formulation of their argument
[11].
The action (3.1.12) possesses two global U(1) pseudo-symmetries, namely a U(1)Φ flavor
symmetry and the U(1)R R-symmetry [12]. The charge assignment are given in Table 3.1.
In particular, the couplings κ1 and κ2 are neutral under both the symmetries and dimensionless,
so we will not indicate them explicitly from now on.
The most general divergent term in the effective action can be written as
d 4
xΓO = λ d 4 xd 4 θ (D 2 ) γ ( ¯ D 2 ) δ (Dα∂ α ˙α D¯ ˙α) η ζ U ρ Φ α Φ¯ β
(3.2.16)
with γ, δ, η, ζ, ρ, α, β non-negative integers. Every derivative is intended as acting on a
(spurion) superfield, and we can also use the commutation relations to lower the number of
spinorial derivatives when necessary. The coefficient λ has dimension d and charges R and S
under the U(1)R and U(1)Φ respectively, and takes the form
λ ∼ Λ d g x−R ¯g x m
Λ
y y+ ¯m
Λ
S−3R
2
λ ω2
2
(3.2.17)
where Λ is an ultraviolet momentum cutoff. Note that the independence on λ1 is fixed by the
fact that we cannot form a 1PI connected diagram from U(D 2 Φ. MOreover, ω2 ≤ ρ because
λ2 only appears coupled to U.
The constraints that (3.2.16) has dimension 4 and zero charge translate into
20
d = 2 + 4ρ − α − β − γ − δ − 2η − 2ζ
R = β − α + 2γ − 2δ − 4ρ
S = β − α. (3.2.18)