Perturbative and non-perturbative infrared behavior of ...

3. The Wess-Zumino model
We stress that we are working with the euclidean space where no hermitian conjugation relations
are assumed for fields, masses and couplings. The general strategy to analyze a deformed theory
is to expand the Moyal product between the fields; note that in doing that, one can always neglect
total spacetime derivatives. Moreover, the ∗-product has no effect when acting on a quadratic
term in the action, as it would contribute only for a total fermionic derivative which vanishes by
definition when integrated in the fermionic coordinates.
To deal with a well-defined superspace expression for all the terms in the lagrangian, we
introduce an external constant spurion superfield U = θ2¯ θ2F 2 . The action takes the form
S = d 8 z¯ ΦΦ − m
2
+ g
6
d 8 zU(D 2 Φ) 3
d 6 zΦ 2 − ¯m
2
d 6 ¯z ¯ Φ 2 − g
3
d 6 zΦ 3 − ¯g
3
d 6 ¯z ¯ Φ 3
(3.1.2)
Note that the term in the last line can be equivalently written as d 6 zΦ(D 2 Φ) 2 , but the integrand
is not chiral and it is not clear why it should be expressed as an F-term. Th use of the constant
spurion superfield allows us to employ all the superspace perturbation theory machinery.
By using d 2 θ ... = D 2 (...) and the projection method, we write the component action as
where
S =
d 4
x φ¯ φ + F ¯ F − GF − ¯ G ¯ F + g
6 C2F 3
+ψ α i∂ ˙α α ¯ ψ ˙α − m
2 ψα ψα − ¯m
2 ¯ ψ ˙α ¯ ψ ˙α − gφψ α ψα − ¯g ¯ φ ¯ ψ ˙α ¯ ψ ˙α
G = mφ + gφ 2
¯G = ¯m ¯ φ + ¯g ¯ φ 2
The auxilary fields F and ¯ F satisfy the algebraic equations of motion (EOM)
F = ¯ G , ¯ F = G − g
2 C2 F 2 = G − g
2 C2 ¯ G 2
(3.1.3)
(3.1.4)
(3.1.5)
To compute quantum perturbative corrections it is convenient to perform the quantumbackground
splitting Φ → Φ + Φq and integrating over the quantum fluctuations Φq. The
expansion produces the ordinary cubic vertices and two vertices involving the spurion field U
and depicted in Fig. 3.1.
The scalar propagators are
16
〈Φ ¯ Φ〉 =
〈ΦΦ〉 = −
〈 ¯ Φ ¯ Φ〉 = −
1
p 2 + m ¯m δ(4) (θ − θ ′ )
¯mD 2
p 2 (p 2 + m ¯m) δ(4) (θ − θ ′ )
m ¯ D 2
p 2 (p 2 + m ¯m) δ(4) (θ − θ ′ ) (3.1.6)