N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

30 Chapter 2. The Vector-Tensor Multiplet
A superfield redefinition induces changes of the component fields. If we define the
components of ˆ L similar to those of L in eq. (2.25), then one has
ψ i = L ′ ˆ ψ i + i∂L λ i
(2.55)
according to eq. (2.53). From this we readily obtain also the relations for W µ and Gµν,
W µ = L ′ ˆ W µ − 1
2 L′′ ˆ ψ i σ µ ˆ¯ ψi − 1
2 ¯ ∂∂L λ i σ µ¯ λi − i
2 (∂L′ λ i σ µ ˆ¯ ψi − ¯ ∂L ′ ˆ ψ i σ µ¯ λi) (2.56)
Gµν = L ′ ˆ Gµν + (∂L + ¯ ∂L)Fµν − i(∂L − ¯ ∂L) ˜ Fµν − 1
2 L′′ ( ˆ ψ i σµν ˆ ψi + ˆ¯ ψ i ¯σµν ˆ¯ ψi)
+ 1
2 (∂2 L λ i σµνλi + ¯ ∂ 2 L ¯ λ i ¯σµν ¯ λi) − i(∂L ′ λ i σµν ˆ ψi − ¯ ∂L ′ ˆ¯ ψ i ¯σµν ¯ λi) ,
while the auxiliary field simply transforms as U = L ′ Û.
2.3.1 Invariant Actions
(2.57)
Once a set of consistent constraints has been found, the construction of a linear superfield
is the crucial step towards an invariant action. Similar to the derivation of the
constraints themselves we start in full generality from an Ansatz for the pre-Lagrangian,
L ij = α D i L D j L + ¯α ¯ D i L ¯ D j L + β D (i Z D j) L + ¯ β ¯ D (i ¯ Z ¯ D j) L
+ γ D i D j Z + δ D i Z D j Z + ¯ δ ¯ D i ¯ Z ¯ D j ¯ Z ,
(2.58)
with γ real. The coefficients are again functions of L, Z and ¯ Z. Whereas reality
and symmetry in ij have already been taken into account, the coefficients are further
constrained by the requirement
D (i
αL jk) = 0 .
Again this yields a set of differential equations analogous to the evaluation of the
consistency condition (2.43). They read
0 = ∂Lγ − 1β
− αC
2
0 = ∂γ − δ − 1
2βC 0 = ∂L ¯α − αG + 2¯αc
0 = ∂ ¯α − 1βG
+ 2¯αa
2
0 = ∂α − 1
2∂Lβ − 1
2
0 = ∂L ¯ β − αB + 2¯αā + ¯ βc
0 = ∂ ¯ β − 1
2 βB + 2¯αb + ¯ βa
0 = ∂β − 2∂Lδ + 2αD − 1
2 βA
0 = ∂L ¯ δ − αE + ¯ βā
0 = ∂ ¯ δ − 1
2 βE + ¯ βb ,
1 βF + αA (2.59)
2
and for given functions A, B, etc. determine the unknown coefficients α, β, etc.