N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

14 Chapter 1. Gauging the Central Charge
We observe that the terms in the second line exactly cancel those in the first which
involve a gauge potential, thereby reducing the covariant derivatives to partial ones.
All that remains is a Lagrangian of (at least classically) free fields,
L0 = ∂ µ ¯ϕi ∂µϕ i − i ↔
χσ
µ
∂µ
2
¯χ ↔
µ
+ ψσ ∂µ ¯ ψ + EF i Fi
¯ , (1.60)
where a total derivative has been dropped. Now consider the second linear superfield
L ij m. It yields the Lagrangian
1
m Lm = iA µ ↔
¯ϕi ∂µϕ i + Aµ(χσ µ ¯χ − ψσ µ ψ) ¯ i
− iE(F ¯ϕi − ϕ i Fi) ¯ − i Yij ¯ϕ i ϕ j
− i( ¯ Z ¯χ ψ¯ − Zχψ) − i ϕ i ( ¯ λi ¯χ − λiψ) + i ¯ϕi(λ i + ¯ λ i ψ) ¯ .
(1.61)
This one involves couplings of the gauge potential to combinations of the scalars and
spinors which are reminiscent of U(1) currents, and indeed we find that the complete
Lagrangian, i.e. the sum L0 + Lm + Lcc,
L = ∇ µ ¯ϕi ∇µϕ i − i ↔
χσ
µ
∇µ
2
¯χ ↔
µ
+ ψσ ∇µ ¯ ψ + E |F i + imϕ i | 2
− m 2 |Z| 2 ¯ϕiϕ i − im( ¯ Z ¯χ ψ¯ − Zχψ) − im Yij ¯ϕ i ϕ j
− im ϕ i ( ¯ λi ¯χ − λiψ) + im ¯ϕi(λ iχ + ¯ λ i ¯ ψ) + Lcc ,
(1.62)
describes nothing but (a special kind of) N = 2 supersymmetric electrodynamics. Here
the operator ∇µ is defined by
∇µ
⎛
ϕ
⎝
i
⎞
⎛
ϕ
χ ⎠ = (∂µ − imAµ) ⎝
¯ψ
i
⎞
χ ⎠ .
¯ψ
(1.63)
Hence, on-shell the gauged central charge generates just local U(1) transformations
with an electric charge that is given by m (or rather mgz after a rescaling Aµ → gzAµ).
This may also be seen from the equation of motion for the auxiliary scalars ¯ Fi (the
relation ≈ denotes on-shell equality),
δL
δ ¯ Fi
= E(F i + imϕ i ) ≈ 0 . (1.64)
Since two superfields are equal if the lowest components coincide, we thus have
δzϕ i ≈ −imϕ i
(1.65)
for the full superfield. One may verify that this is in agreement with the eqs. (1.53).
Note that since 〈Z〉 = i, the masses of the “electron” (χ, ¯ ψ) and its superpartners ϕ i
are given by the parameter m (where we assume m ≥ 0),
M (χ, ¯ ψ) = Mϕ = m , (1.66)