JHEP05(2018)102
2 Higher-curvature terms in supergravity
In this section, we review the construction of hi gh er -or de r terms of the Ricci scalar in 4D
N = 1 SUGRA [56] .
2
In this paper, we use the conformal SUGRA formalism, in which there
are conformal symmetry and its SUSY counter par ts in addition to sup e r -Poincar´e symme-
try [
64–67]. In order to fix the extra gauge degree of freedom, we need to intro du ce an un-
physical degrees of freedom called the conformal compensator, which should be in a s uper-
conformal multiplet. In this paper, we adopt a chiral superfield as a compensator superfield,
which leads to the so-call e d old minimal SUGRA after superconformal gauge fixing. We
show the components of supermultiplet, the density formulas, and identities in appendix A.
First, let us show the pure conformal SUGRA action,
S =
−
3
2
S
0
¯
S
0
D
, (2.1)
where S
0
is the chiral compensator with the charges (w, n) = ( 1, 1) in conformal SUGRA
(see Apeendix
A for the definition of the charges), and [···]
D
denotes the D-term density
formula. Taking the pure SUGRA gauge, S
0
=
¯
S
0
= 1, b
µ
= 0, we obtain an action whose
bosonic part takes the form
S =
Z
d
4
x
√
−g
1
2
R − 3|F
S
0
|
2
+ 3A
a
A
a
, (2.2)
where R is the Ricci scalar, F
S
0
is the F-term of S
0
and A
a
is the gauge field of chiral U(1)
A
symmetry, which is a part of superconformal symmetry. The E.O.M. for the auxil i ar y fields
F
S
0
and A
a
can be solved by setting F
S
0
= A
a
= 0, and then we find th e pure SUGRA
action. The action (
2.1) can also be written as
S =
3
2
S
2
0
R
F
, (2.3)
where [···]
F
is the F-term density formula. Here we have used the identity given in (
A.26).
The chiral superfie l d R is the so-called s cal ar curvature superfield, defined by
R ≡
Σ(
¯
S
0
)
S
0
, (2.4)
where Σ is the chiral projection operator. Its components in the pure SUGRA gauge are
given by
R = [Φ , P
L
χ , F ] =
−
¯
F
S
0
, ··· , | F
S
0
|
2
+
1
6
R + A
a
A
a
− i∂
a
A
a
+ ···
, (2.5)
where el l i p se s denote ferm i oni c parts. From this expression, we find that the F-component
of R contains the Ricci scalar .
It has been known that there is no ghost in the system involving R
2
, which is realized as
S =
−
3
2
S
0
¯
S
0
+
α
2
R
¯
R
D
, (2.6)
2
Cosmological application of SUSY Starobinsky model is discussed e.g. in [
61–63].
– 4 –
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