N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

66 Chapter 4. The Nonlinear Case
where for consistency the fermions contained in the composite fields W µ and Gµν have
to be set to zero. The latter depends on the former according to eq. (4.21), so we first
replace Gµν,
− 1
4 GµνGµν + 4AµWν + 2εµνρσ ∂ ρ (LA σ ) =
= − 1
4 V µν Vµν − ˜ V µν ∂µ(LAν) − LWµ ˜ F µν Aν + A µ W ν A[µWν] .
Then the terms linear in W µ cancel, while the bilinear ones combine into W µ KµνW ν
just as for the linear vector-tensor multiplet,
1
LnlinVT ϱL
2 |Z|2∂ µ L ∂µL + EU 2 − 1
2 W µ KµνW ν − 1
4 V µν Vµν − 1
4 L ˜ F µν Vµν
+ 1
12 L2F µν Fµν − 1
6 L2 (Z ¯ Z + ¯ ZZ) − 1
12 L2 Y ij
Yij
+ ϱ
6 ∂µ
3 µν
2L F Aν − 3L 2
V ˜ µν
Aν .
(4.47)
Again the nonpolynomial interactions arise from inverting Kµν. Using eqs. (3.38) and
(4.29), the substitution of W µ gives
− ϱ
2 L W µ KµνW ν = − ϱ
2L (LKµρWρ) (K −1 )µν (LK νσ Wσ)
= − ϱ µ 1 H − 2 2LE
˜ V µν Vν + 1
2L2F ˜µν Aν + (LV µν + Π µν )Aν
ϱ
+
2LE|Z| 2
AµH µ − 1
2Aµ ˜ V µν 2 Vν .
2
(4.48)
At last, let us neglect also fluctuations of the scalars around their background values
〈Z〉 = i and 〈L〉 = 1/ϱ. Then only the gauge potentials Vµ, Bµν and Aµ remain, and
after rescaling
Aµ → gzAµ , Bµν → Bµν/ϱ ,
such that both fields have canonical dimension 1, we find 2 (dropping a total derivative)
L = − 1
4 V µν Vµν − 1 2
1 − g
4
z/3ϱ 2 F µν Fµν
− 1 µ 1 H − 2 2E
ϱ ˜ V µν Vν + gzV µν Aν + 1
2ϱ−1g 2 z ˜ F µν Aν
AµH µ − 1
2ϱAµ ˜ V µν 2 Vν .
+ g2 z
2E
2
(4.49)
Apart from the normalisation of Aµ, this Lagrangian follows from the one in eq. (3.64),
when in the latter we make the substitution
H µ → H µ − 1
2 ϱ ˜ V µν Vν + 1
2 ϱ−1 g 2 z ˜ F µν Aν , (4.50)
2 Evidently, positivity of the kinetic energies requires 3ϱ 2 > g 2 z.