Ghostbusters in f (R) supergravity
HJE
Ghostbusters in f (R) supergravity
Toshiaki Fujimori 0 2
Muneto Nitta 0 2
Keisuke Ohashi 0 2
Yusuke Yamada 0 1
0 382 Via Pueblo Mall , Stanford, CA 94305 , U.S.A
1 Stanford Institute for Theoretical Physics and Department of Physics, Stanford University
2 Department of Physics & Research and Education Center for Natural Sciences, Keio University
f (R) supergravity is known to contain a ghost mode associated with higherderivative terms if it contains Rn with n greater than two. We remove the ghost in f (R) supergravity by introducing auxiliary gauge field to absorb the ghost. We dub this method as the ghostbuster mechanism [1]. We show that the mechanism removes the ghost supermultiplet but also terms including Rn with n ≥ 3, after integrating out auxiliary degrees of freedom. For pure supergravity case, there appears an instability in the resultant scalar potential. We then show that the instability of the scalar potential can be cured by introducing matter couplings in such a way that the system has a stable potential.
Supergravity Models; Supersymmetric Effective Theories
Highercurvature terms in supergravity
Ghostbuster in f (R) supergravity
3.1
Ghostbuster in pure f (R) supergravity model
3.2 Instability of scalar potential
Stable ghostbuster model with extra matter
4.1
4.2
Preliminary
Example of matter coupled f (R) supergravity
Ghostbuster mechanism from highercurvature SUGRA viewpoint
Conclusion
A Superconformal tensor calculus
1 Introduction
2
3
4
5
6
1
Introduction
Higherorder derivative interactions naturally appear in effective field theories. In
particular, in the system with gravity, we need to take into account such terms since
various higherorder corrections can be relevant to the dynamics. However, higherderivative
interactions often lead to the socalled Ostrogradski instability [2, 3]: higherderivative
interactions give additional degrees of freedom which makes the Hamiltonian unbounded
from below, and hence the system shows an instability. If such a ghost mode appears,
one should regard the system as an effective theory which is valid only below the energy
scale of the mass of the ghost mode, otherwise the system loses the unitarity. In a class of
ghostfree higherderivative interactions, one does not come across with such an instability
problem. In the case of a system with a single scalar and a tensor, the Horndeski class [4, 5]
of interactions are free from ghosts. In this class of interactions, the equations of motion
(E.O.M) are at most the second order differential equations, and no additional degree of
freedom shows up. In general, one may ask the following question: among many possible
higherorder derivative terms, what kind of structure gives us ghostfree interactions? For
example, in the socalled Galileon models [6], Galileon scalar fields can be understood as
the Goldstone mode of translation symmetry in extra dimensions, and the action is made
out of ghostfree derivative terms. Therefore, one can say that the hidden translation
symmetry controls the higherderivative interactions so that there appear no new degrees of
– 1 –
can extend this model to the system with an arbitrary function of the Ricci scalar, called
the f (R)gravity model [11] (see also refs. [12, 13] for review), which is also dual to a
scalartensor system, and therefore free from the ghost instability.
Higherderivative interactions were also studied in supersymmetric (SUSY) theories,
both for global SUSY and supergravity (SUGRA). In SUSY cases, there is another problem
called the auxiliary field problem: spacetime derivatives may act in general on SUSY
auxiliary fields (F and D for chiral and vector multiplets, respectively) in the offshell
superfield formulation. Then, they become dynamical and so one cannot eliminate them
by their E.O.M [14, 15]. The auxiliary field problem and the higherderivative ghosts
usually come up together [16, 17]. In four dimensional (4D) N = 1 SUSY theories, a
classification of higherderivative terms free from ghosts and the auxiliary field problem
was given for chiral superfields [18–21] as well as for vector superfields [22]. Such
higherderivative interactions of chiral superfields were applied to lowenergy effective theory [23–
26] (see also [27]), coupling to SUGRA [20, 28], Galileons [19], ghost condensation [21], a
DiracBornInfeld (DBI) inflation [29], flattening of the inflaton potential [30, 31], a (baby)
Skyrme model [32–40], other BPS solitons [34, 35, 41, 42], and modulated vacua [43, 44],
while higherderivative interactions of vector superfields were applied to the DBI action [45–
47], SUGRA coupling [45, 48–50], SUSY EulerHeisenberg action [17, 28, 51, 52], and
nonlinear selfdual actions [48, 49, 53–55].
On the other hand, higherderivative interaction of gravity multiplets were studied in
4D N = 1 SUGRA. In ref. [56], Cecotti constructed the higherorder terms of the Ricci
scalar in the old minimal supergravity formulation and showed that at least one ghost
superfield appears if we have Rn (n ≥ 3) terms in the system. It is possible to avoid the
ghost by some modifications of the system. In [57], the socalled nilpotent constraint on
the Ricci scalar multiplet, which removes a scalar field in the multiplet, is considered. Due
to the absence of the scalar, the bosonic ghost is absent in the spectrum of the system.
This mechanism has been applied to various highercurvature models in SUGRA [58]. The
nilpotent constraint R
2 = 0, however, is an effective description of a brokenSUSY system.
If the linearly realized SUSY is restored in a higher energy regime, the ghost mode would
show up.1 As another approach, in [59] the authors considered a deformation of the ghost
kinetic term by introducing an additional K¨ahler potential term. It is shown that the
resultant ghostfree system is equivalent to the matter coupled f (R) SUGRA.
1The nilpotent condition on a chiral superfield Φ has two solutions. A nontrivial solution is φ = Fψψφ
where φ, ψ and F φ are scalar, Weyl spinor, and auxiliary scalar components of Φ. Obviously, this solution
is welldefined for F φ 6= 0, that is, SUSY should be spontaneously broken.
– 2 –
Meanwhile, in our previous work [1], we proposed a simple method to remove a ghost
mode in 4D N = 1 SUSY chiral multiplets [16, 17], which we dubbed “ghostbuster
mechanism.”
We gauge a U(1) symmetry by introducing a nondynamical gauge superfield
without kinetic term to the higherderivative system with assigning charges on chiral
superfields properly in order for the gauge field to absorb the ghost. Namely, due to the
gauge degree of freedom, the ghost in the system is removed by the U(1) gauge fixing. In
this class of models, a hidden local symmetry plays a key role in the ghostbuster
mechanism. Actually, before this work, esentially the same technique is used for superconformal
symmetry in the conformal SUGRA formalism: the conformal SUGRA has one ghostlike
degree of freedom called a compensator. Such a degree of freedom is removed by the
superconformal gauge fixing, whereas in the ghostbuster mechanism, the hidden local U(1)
gauge fixing removes the ghost associated with higherderivatives. Therefore, in SUGRA
models, one may understand the higherderivative ghost as a second compensator for the
system with the superconformal symmetry × hidden local U(1) symmetry.
In this paper, we apply the ghostbuster mechanism to remove the ghost in the f (R)
SUGRA system. Interestingly, the hidden U(1) symmetry required for the mechanism
can be understood as the gauged Rsymmetry, since the gravitational superfield should be
gauged under the U(1) symmetry. The U(1) charge assignment is uniquely determined, and
therefore, naively one cannot expect a ghost mode cancelation a priori. As we will show,
a wouldbe ghost superfield has a gauge charge and can be nicely removed by the gauge
fixing of the U(1) symmetry. As a price of this achievement, however, the resultant system
generically has an unstable scalar potential in a pure SUGRA case. Such an unstable scalar
potential can be cured by various modifications. As an example we propose a model with
a matter chiral superfield. We will find that such a deformation leads to a healthy model
of SUGRA without either ghosts or instabilities of the scalar potential.
One will easily find how the ghost supermultiplet is eliminated from the dual
mattercoupled SUGRA viewpoint. We also address the same question in the highercurvature
SUGRA system. We find that, after integrating out the auxiliary vector superfield for the
mechanism, the scalar curvature terms including Rn with n ≥ 3 disappear, and the
resultant system has linear and quadratic terms in R. However, the R + R2 SUGRA system has
couplings completely different from that proposed in [56]. This observation means that,
despite the disappearance of higher scalar curvatures in the final form, the highercurvature
deformation in the original action gives a physical consequence even after applying the
ghostbuster mechanism.
This paper is organized as follows. In section 2, we briefly review the highercurvature
SUGRA models and its dual description. In particular, one finds that once the SUSY
version of the higher order Ricci scalar term Rn (n ≥ 3) is included in the old minimal
SUGRA formulation, there appears at least one ghost chiral superfield. We apply the
ghostbuster mechanism to the highercurvature SUGRA in section 3. We will see that
although the ghost superfield can be removed by the mechanism, the resultant system has
,
where S0 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 Dterm density
formula. Taking the pure SUGRA gauge, S0 = S¯0 = 1, bμ = 0, we obtain an action whose
bosonic part takes the form
where R is the Ricci scalar, F S0 is the Fterm of S0 and Aa is the gauge field of chiral U(1)A
symmetry, which is a part of superconformal symmetry. The E.O.M. for the auxiliary fields
F S0 and Aa can be solved by setting F S0 = Aa = 0, and then we find the pure SUGRA
action. The action (2.1) can also be written as
In this section, we review the construction of higherorder 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 counterparts in addition to superPoincar´e
symmetry [64–67]. In order to fix the extra gauge degree of freedom, we need to introduce an
unphysical degrees of freedom called the conformal compensator, which should be in a
superconformal multiplet. In this paper, we adopt a chiral superfield as a compensator superfield,
which leads to the socalled 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,
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
where [· · · ]F is the Fterm density formula. Here we have used the identity given in (A.26).
The chiral superfield R is the socalled scalar curvature superfield, defined by
where Σ is the chiral projection operator. Its components in the pure SUGRA gauge are
given by
R = [Φ , PLχ , F ] =
−F¯S0 , · · · , F S0 2 +
R + AaA
a
− i∂aAa + · · · ,
1
6
where ellipses denote fermionic parts. From this expression, we find that the Fcomponent
of R contains the Ricci scalar.
It has been known that there is no ghost in the system involving R2, which is realized as
S =
Σ(S¯0) ,
– 4 –
2Cosmological application of SUSY Starobinsky model is discussed e.g. in [61–63].
where α is a real constant. The bosonic part of this action after the superconformal gauge
fixing is
SB =
Z
d x
4 √
−g
" R
2
+
α
36
+
αR
6
R − 3F S0 2
2
 − αDaF S0 DaF¯S0 + 3AaAa + α(∂aAa)2

F S0 2 + 2AaAa + α 
F S0 2 + AaAa 2 ,
#
(2.7)
where Da represents the covariant derivative, DaS0 = (∂a − iAa)S0 = −iAa, DaF S0 =
(∂a + 2iAa)F S0 . The Lagrangian has the quadratic Ricci scalar term 3α6 R2 and also the
nonminimal couplings between F S0 , Aa and R. In this system, there exist four real
massive modes ϕi with the common mass m2 = 3/α in the fluctuations around the vacuum
gμν = ημν and F S0 = Aa = 0:
We stress that, as is often the case with SUSY higher derivative models, the auxiliary fields
have their kinetic terms and hence they are dynamical degrees of freedom in the presence
of the higherderivative term.
Next, let us consider a SUGRA system with Rn, n ≥ 3 along the line of refs. [56,
57, 68]. As we discussed in the previous section, R superfield has the Ricci scalar in its
Fcomponent. Using the chiral projection operator Σ, one can obtain the superfield Σ(R¯ )
which has R in the lowest component:
1
− 6
1
6
– 5 –
Σ(R¯ ) =
R − F S0 2
 − AaAa
− i∂aAa + · · · , · · · ,
RF S0 + ∂a2 + i∂aAa
− AaAa F S0 + · · · ,
S =
where we have shown only the relevant part. With this superfield Σ(R¯ ), one can construct
an action involving arbitrary functions of R, i.e. f (R) gravity models in SUGRA. Here we
consider the action of the form
3
− 2 S0S¯0Ω
¯
SR0 , SR¯0 ,
0
ΣS(R2¯ ) , Σ¯ (R)
S¯2
0
D
where Ω is an arbitrary real function and F is an arbitrary holomorphic function. If we
chose Ω = 0, F (S, X) = S(3 − αX)/2, then this action reduces to (2.6) since
R ,
S0
Σ(R¯ )
S2
0
F
=
α
(2.12)
(2.9)
(2.10)
(2.11)
where the subscripts on the functions denote the differentiations with respect to the scalar
fields.
Such SUSY higherderivative terms have derivative interactions of auxiliary fields, and
the interactions make the auxiliary fields dynamical as
d x
4 √
−g
12
1 gμν ∂μR ∂ν R ΩXX¯ + ∂μ2F S0
FX + h.c. + · · ·
In this system, in addition to the scalar degree of freedom from the derivative terms of
the Riccicurvature, the higherderivative terms of the “dynamical” auxiliary field F S0
give rise to multiple scalar degrees of freedom, some of which are ghostlike. If we choose
Ω(S, S¯, X, X¯ ) = SS¯Ω˜ (X, X¯ ), F (S, X) = SF˜(X), and set F S0 = 0 identically as is done
by imposing the nilpotent condition R
2 = 0 in ref. [57], the above terms vanish and no
ghost seems to appear. Without such a condition, however, the appearance of ghost is
unavoidable as is clearly shown in the following.
The present system is also equivalent to a standard SUGRA model coupled to matter
superfields. As in the previous section, we use Lagrange multiplier suerfields, and rewrite
the action (2.10) as
identity (A.26), we can also obtain the dual action
′
S =
3
− 2 S0S¯0 T + T¯ + Y S¯ + Y¯ S + Ω(S, S¯, X, X¯ )
D
+ S03 (F (S, X) − 3T S − 3XY )
This is a standard SUGRA system with the following K¨ahler and superpotentials,
K = −3 log T + T¯ + Y S¯ + Y¯ S + Ω(S, S¯, X, X¯ ) ,
W = F (S, X) − 3T S − 3XY.
Let us show the existence of a ghost mode. The K¨ahler metric of the {S, Y } sector
takes the form,
where A = T + T¯ + Y S¯ + Y¯ S + Ω(S, S¯, X, X¯ ). The determinant of this sub matrix has
negative determinant, and this K¨ahler metric has one negative eigenvalue corresponding
to a ghost. Thus, the f (R) SUGRA model has one ghost mode in general.
KIJ¯ =
KSS¯ − A
1
− A
0
1 !
,
– 6 –
Note that X becomes an auxiliary superfield if Ω = Ω(S, S¯) is independent of X. Even
in such a case, the system has highercurvature terms in the F (S, X) term in (2.12). The
reduced dual system is described by
K = −3 log T + T¯ + Y S¯ + Y¯ S + Ω(S, S¯) ,
W = g(S, Y ) − 3T S,
where g(S, Y ) = [F − XFX ]X=X(S,Y ) and X(S, Y ) is a solution of FX − 3Y = 0.3 This
reduction does not change the above discussion, and hence a ghost mode appears in this
system as well.
HJEP05(218)
Ghostbuster in f (R) supergravity
In this section, we consider the elimination of the ghost superfield along the line of ref. [1].
To eliminate the ghost superfield, one needs to introduce a gauge redundancy, by which
one of the degrees of freedom is removed. In the f (R) SUGRA discussed above, all the
superfields R, Σ(R¯) are expressed in terms of S0 with the SUSY derivative operators.
Hence, once we introduce a vector superfield VR for a U(1) gauge symmetry and assign the
charge to S0 so that it transforms as
(2.19)
(3.1)
(3.2)
S0 → eΛS0,
VR → VR − Λ − Λ¯,
the transformation law of R and Σ(R¯) are automatically determined as
Rg ≡
Σ(S¯0eVR)
S0
→ e−2Λ
Rg, Σg(R¯) ≡ Σ(R¯ge−2VR) → e2ΛΣg(R¯),
where the chiral projection Σ needs to be modified so that the operations is covariant under
the gauge symmetry. In the rest of this section, we omit the suffix g attached to Rg, Σg.
Interestingly, the U(1) gauge symmetry under which the compensator is charged becomes
a gauged Rsymmetry [69]. We call it a U(1)R symmetry in the following discussion. Here,
however, we do not introduce a kinetic term for VR and thus the vector superfield VR is an
auxiliary superfield, which should be written as a composite field consisting of curvature
superfields R and Σ(R¯).
3.1
Ghostbuster in pure f (R) supergravity model
Let us introduce a U(1)R gauge symmetry under which S0 has charge cS0 = 1. Since the
chiral superfield R = Σ(S¯0)/S0, the charge of R is determined as cR = −2. Analogously,
we find that cΣ(R¯) = 2. Then the gauged extension of the system (2.10) with Ω = Ω(S, S¯)
is described by the action
3
S = − 2 S0 eVRS¯0 Ω
S0 S0
R , R¯¯ e−3VR
D
R ,
(3.3)
3Here we assume that the equation FX = FX(X) can be solved for X (e.g. F ∝ SXn−1 with n ≥ 3).
Constant and linear terms in F merely rescale the R and R2 terms respectively.
– 7 –
where Ω should be gauge invariant and F should have gauge charge cF = −3 in total.
Hence F should take the form
F
S0
Σ(R¯)
Σ(R¯)
S0
To discuss the ghost elimination, it is useful to consider the dual system as in the
nongauged case (2.15). The dual system of the gauged model is described by
′
F
#
F
,
where the gauge charges of T, S, X, Y are (cT , cS, cX , cY ) = (0 , −3 , 0 , −3). Similarly to
the nongauged case, we can rewrite this action as
− 2 S0 eVRS¯0 T + T¯ + Y S¯e−3VR + Y¯ e−3VRS + Ω(S, S¯e−3VR)
′
S =
"
"
3
+ 3S03 F˜(X)S − T S − XY
The variation of VR gives the following E.O.M for VR
(γ + T + T¯)eVR − 2 Y S¯ + SY¯ − hSS¯ e−2VR = 0.
This equation can be algebraically solved in terms of VR as
e−3VR =
γ + T + T¯
2 Y S¯ + SY¯ − hSS¯ .
– 8 –
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
For simplicity, in the following discussion, we choose the function Ω = γ − hSS¯e−3VR,
where γ is a real constant. Note that one can perform the following procedure with a more
general form of Ω in a similar way. Then we obtain
"
3
"
− 3S03 F˜(X)S + T S + XY
S =
− 2 S0 eVRS¯0 γ + T + T¯ + Y S¯ + Y¯ S − hSS¯ e−3VR
We stress that the U(1)R charges of (S, Y ) are automatically determined to be nonzero.
This is a nontrivial and important nature of the f (R) SUGRA model since the ghostbuster
mechanism does not work if S and Y , either of which corresponds to the ghost mode, did
not have the U(1)R charges.
#
F
.
#
F
Substituting this solution to the action, one finds
,
where we have rescaled S0 as S0 → 21/3/√3 S0. Thus, starting from the modified
highercurvature action (3.3), we find the dual mattercoupled system (3.10). After partial gauge
fixings of superconformal symmetry,4 this system becomes Poincar´e SUGRA with the
fol
This system is invariant under the U(1)R gauge transformation {S, Y } → {
Therefore, if the lowest component of Y takes a nonzero value, we can fix the U(1)R gauge
eΛS, eΛY }.
by setting Y = 1. Then, after a redefinition S → S + h1 , we obtain
2
2
3
W = − √
2
3
F˜(X)(S + 1/h) + T (S + 1/h) + X .
If S 6= 0, we can also fix the gauge by setting S = 1. Then we find
2
3
K = −2 log(γ + T + T¯) − log(Y + Y¯ − h),
W = − √
F˜(X) + T + XY .
Except for the two points S = 0 (Y = ∞), Y = 0 (S = ∞), the above two descriptions are
equivalent and related by a coordinate transformation between S and Y . In both cases,
all the eigenvalues of the K¨ahler metric are obviously positive. Therefore, we have shown
that the ghost mode is eliminated by our ghostbuster mechanism. Note that X is an
auxiliary field in this setup, and we need to solve the E.O.M for X to obtain the physical
superpotential.
We stress that the elimination of the ghost mode by the ghostbuster mechanism in this
highercurvature system is nontrivial since we do not have any choice of the charge
assignment to the superfields. As we have seen above, the wouldbe ghost modes have charges
under U(1)R, which enables us to remove the ghost mode by the gauge degree of freedom.
4More specifically, we fix dilatiation, chiral U(1) symmetry, SSUSY, and conformal boost, so that
Poincare SUSY remains in the resultant system. The detailed procedure of superconformal gauge fixing is
discussed e.g. in [60].
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
In this section, we analyze the scalar potential of the ghostfree system derived in the
previous section. The Fterm scalar potential in the Poincar´e SUGRA is given by
V = eK hKAB¯ (WA + KAW )(W¯ B¯ + KB¯ W¯ ) − 3W 2i .
If we choose the gauge fixing condition S = 1, Im T appears only in W due to the shift
symmetry of Im T in the K¨ahler potential, and hence the mass of Im T is given by
The K¨ahler potential in eq. (3.15) has the property called the noscale relation
Since W ∝ T + XY , the potential has the following linear term of Im T
mI2m T ∝ eK (KAB¯ KAKB¯ − 3).
KAB¯ KAKB¯ = 3.
Im(KB¯ KB¯AWA) Im T.
(3.18)
the K¨ahler potential in eq. (3.15), we find that the coefficient of the linear term is given by
Im(KB¯ KB¯AWA) =
√ (Y + Y¯ − h) ImX.
4
3
Note that the nondynamical field X becomes a function of Y after solving its E.O.M.
Im T has only a mass term ∼ h
Y + Y¯ − hi(XY Y + XY Y¯ )Im T , where XY ≡ h∂Y (ImX)i.
Unfortunately, this “offdiagonal” contribution in the mass matrix leads to a tachyonic
mode.5 This instability cannot be cured by any higherorder terms since Im T appears
only in the term (3.20). Therefore, ImX 6= 0 makes Im T unstable and even if there is the
local minimum in ImX = Im T = 0, that point cannot be a local minimum, but must be
a saddle point. We conclude that although the instability caused by ghost mode is absent
thanks to the ghostbuster mechanism, the pure highercurvature action has an unstable
scalar potential, which does not have any stable SUSY minimum. In the next section, we
consider an extension of our model to improve this point.
4
4.1
Preliminary
Stable ghostbuster model with extra matter
As we discussed in the previous section, the scalar potential of our minimal model has no
stable SUSY minimum. One may improve such a situation by various types of
modifications. Here we take a relatively simple way; we introduce an additional matter field Z
so that the coupling between the gravitational sector and the additional sector stabilizes
5In general, hY + Y¯ − hi should be nonzero since the K¨ahler potential has − log(Y + Y¯ − h) and diverges
for hY + Y¯ − hi = 0.
S0 eVR S¯0 Ω(S, S¯e−3VR , Z, Z¯)
+ S03SF˜(X)
#
F
"
which can be rewritten as
0 F˜(X)S − 3T SZ − 3XY
3
#
F
For simplicity, let us choose the function as
Ω = γ − g(Z, Z¯) − h(Z, Z¯) SS¯ e−3VR .
We can also change the definition of X as
S = R
S0
→
SZ = R .
S0
X =
with an arbitrary function k(Z, Z¯). Note that if we chose k(Z, Z¯) = Z, then we obtain
the same unstable model as in section 3 with the redefinition S → S′ = SZ. Therefore,
k(Z, Z¯) should have a constant term around the minimum of Z, i.e. k(hZi, hZ¯i) ≡ c 6= 0.
Under this modification, the dual system is given by
the potential.6 Let us assume that Z carries no U(1)R charge so that the
superpotential W contains T Z term in the S = 1 gauge. Then it is possible to introduce Z in the
superpotential in such a way that the constraint for S is modified as
′
After solving the E.O.M for VR, we find the following K¨ahler potential and superpotential
K = −2 log hγ + T + T¯ − g(Z, Z¯)i − log hY k¯(Z¯) + Y¯ k(Z) − h(Z, Z¯)i,
W = √
2
3 31 F˜(X) − (T Z + XY ) ,
in the S = 1 gauge.
6Even in the R2 model, the deformation of scalar potential of T corresponding to the scalaron superfield
requires an additional degree of freedom in the dual highercurvature SUGRA action [70].
4.2
Let us discuss a simple example by setting the functions as
The corresponding K¨ahler potential is given by
k(Z) = c + Z,
Ω = γ + (β − bZZ¯) SS¯ e−3VR .
K = −2 log ω1 − log ω2,
ω1 ≡ γ + T + T¯,
ω2 ≡ β + Y¯ (c + Z) + c.c. − bZZ¯,
+ S02(S1 − Rg)XG(X)
F
where both ω1 and ω2 are required to be positive so that there exists a solution of the
E.O.M. for VR and the condition eK > 0. The eigenvalues {λi  i = 1, 2, 3} of the K¨ahler
metric KAB¯ are given by
λ1 =
2
ω2
1
,
λ2 + λ3 = ∂Y ω22 + ∂Z ω22 + b ω2
,
λ2λ3 = bc2 − β
.
ω2
2
ω3
2
Furthermore, by choosing the function F˜ so that F˜(0) = 0, F˜′(0) = 0, we find a SUSY
vacuum satisfying WA = W = 0 at X = Y = T = S = 0, which is guaranteed to be stable.
Therefore, there exists the SUSY vacuum with a positive definite metric if and only if
γ = ω1vac > 0,
β = ω2vac > 0, b >
c 6= 0.
β
c2
,
When these conditions are satisfied, there exist no ghost anywhere in the region M
=
{T, Y, Z  ω1 > 0 , ω2 > 0} and the boundary ∂M is geodesically infinitely far away from
the SUSY vacuum.
5
Ghostbuster mechanism from highercurvature SUGRA viewpoint
In this section, we discuss how the ghostbuster mechanism works in the highercurvature
frame. As we have seen in previous two sections, the ghost supermultiplet is eliminated in
both pure and mattercoupled highercurvature systems.
Let us consider the original action for f (R) gravity before taking the dual
transformation. For concreteness of the discussion, we take the simplest model with an additional
matter superfield in eq. (4.8). The same conclusion follows even in the absence of an
additional matter. The highercurvature action can be obtained by solving E.O.M. for T and
Y and imposing the constraints for S and X. Here we introduce S1 ≡ cS0S + Rg as an
extra matter and solve the modified constraint (4.1) for Z. After introducing the quadratic
term of X, the original action takes the form
′
S =
3
3β
− 2 γS02eVR − 2c2 S1 − Rg2e−2VR +
3
2 aS0X2eVR +
3
2 bRg2e−2VR
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
D
(5.1)
with F˜(X) ≡ cXG(X) and
Rg =
Σ(S¯0eVR )
S0
,
X =
Σ(S¯1e−2VR )
S2
0
,
(5.2)
where a and b are real (positive) parameters. Note that X now does not have the Ricci
scalar in the lowest component but a higherderivative superfield made out of S1. This
means that the higherderivative term of Rg is now replaced by that of S1, and hence the
highercurvature term does not show up. By expanding the action explicitly, one can check
that this action has Ricci scalar terms up to the quadratic order. We note that, however,
this does not lead to the conclusion that the ghost is removed by the additional matter:
since there still exist higherderivative terms of S1, the ghost mode can arise from such
terms. One may also confirm that the absence of the higher curvature terms Rn (n ≥ 3)
is not an artifact of field redefinition. We can show that in this specific matter coupled
model, the highercurvature terms exist only in the offshell action before substituting the
solution of the E.O.M for the auxiliary field in VR.
We stress that this conclusion does not mean that the highercurvature modification
is removed by the ghostbuster mechanism. As we claimed above, the resultant system
has scalar curvature terms only up to the quadratic order, as the simplest Cecotti model
does [56]. However, the coupling of the resultant system is completely different from the
Cecotti model. In our dual matter coupled system in section 4.2, K¨ahler potential (4.9)
takes the form
K ∼ −2 log(T + T¯) − log(Y + Y¯ + · · · ),
whereas, in the Cecotti model, it can be written as
K = −3 log(Tc + T¯c + · · · ),
where T, Y and Tc are chiral superfields. The difference of the K¨ahler potentials leads
to a different moduli space geometry. Interestingly, all T, Y and Tc have the hyperbolic
geometry structure, which is applicable to the socalled inflationary αattractors [71, 72].
In the αattractor inflation, we take the moduli space K = −3α log(Φ + Φ¯ ) for an inflaton
superfield Φ, and the value of the parameter α has a relation to the tensor to scalar ratio r
aαs =r =13 1aN2nα2d, w32,hewrheeNreaiss tthhee nCuemcobtetri omf oedfeolldhiansgsαa=tth1.e hIforwizeoanpepxliyt.oIunr omuordmelodtoel,inwfleathiaovne,
we would find a value of tensor to scalar ratio r different from that of the Cecotti model.
Therefore, the highercurvature modification has physical consequences even though the
higherorder scalar curvature terms seem to disappear after the ghostbuster mechanism.
Since the construction of the inflation model is beyond the scope of this paper, we leave it
(5.3)
(5.4)
HJEP05(218)
as future work.
6
Conclusion
We have applied the ghost buster method to a highercurvature system of SUGRA. It has
been known that once we introduce a higher scalar curvature multiplet Σ(R¯ ), a ghost mode
generically shows up in the system as we reviewed in section 2. The ghostbuster method
requires a nontrivial U(1) gauge symmetry with a nonpropagating gauge superfield. It turned
out that the required U(1) symmetry should be the gauged Rsymmetry in the case of the
highercurvature system, since the ghost arises from the gravitational superfield. Due to the
uniqueness of the gauge charge assignment, it is nontrivial that if the ghostbuster method is
applicable to remove the ghost. As we have shown in section 3, thanks to the nonzero U(1)
charge of “wouldbe” ghost mode, we can eliminate the ghost mode and obtain a ghostfree
action. However, the resultant ghostfree system turned out to be unstable because of the
scalar potential instability. Such an instability is easily cured by introducing matter fields,
which would be necessary for realistic models. Additional matter superfields can stabilize
the scalar potential if we choose proper couplings between gravity and matter multiplets.
We have also discussed how the ghostbuster mechanism can be seen in the
highercurvature system in section 5. We have found that the higherorder scalar curvature terms
Rn with n ≥ 3 are eliminated in using the mechanism, and the resultant system has the
scalar curvature up to the quadratic order. However, the highercurvature modification
is not completely eliminated by the mechanism. We find moduli space geometry different
from the known R + R2 supergravity [56]. Therefore, despite the absence of f (R) type
interactions in the final form, the SUSY higherorder curvature corrections give physical
differences. In particular, the difference of the moduli space structure might be useful for
constructing inflationary models.
In this work, we did not discuss the elimination of ghosts originated from
higherderivative terms of matter superfields. It is a straightforward extension of our previous
work [1] for global SUSY to SUGRA and is much easier than the highercurvature model
discussed in this paper, since the U(1) charge assignment is not unique for matter
higherderivative models. Since the higherderivatives of matter fields in SUGRA requires the
compensator S0, it would be interesting to assign the U(1) charge to the compensator as
well, i.e. we can use U(1) Rsymmetry for the ghostbuster mechanism as with the
highercurvature case, which is only possible for the SUGRA case.
Let us mention the applicability of our mechanism to the other SUGRA formulations,
where the auxiliary fields in the gravity multiplet are different. Our mechanism is not
applicable for the socalled new minimal SUGRA formulation [73], since the compensator
is a real linear superfield, which cannot have any U(1) charge. For the nonminimal SUGRA
case, it would be possible to assign a nontrivial U(1) charge to complex linear compensator.
In addition, it is known that the R2 model of nonminimal SUGRA has a ghost mode in
the spectrum, so it is interesting to see if the ghost can be removed by our mechanism.
Acknowledgments
This work is supported by the Ministry of Education, Culture, Sports, Science
(MEXT)Supported Program for the Strategic Research Foundation at Private Universities
“Topological Science” (Grant No. S1511006). The work of M. N. is also supported in part by a
GrantinAid for Scientific Research on Innovative Areas “Topological Materials Science”
(KAKENHI Grant No. 15H05855) from the MEXT of Japan, and by the Japan Society for
the Promotion of Science (JSPS) GrantinAid for Scientific Research (KAKENHI Grant
No. 16H03984). Y. Y. is supported by SITP and by the NSF Grant PHY1720397.
A
Superconformal tensor calculus
Here we give a brief summary of the superconformal formulation. We use the convention
ηab = diag(−1, 1, 1, 1) for the Minkowski metric.
In 4D N = 1 conformal SUGRA, we have the superPoincar´e generators {Pa, MabQα},
and the additional superconformal generators, {D, A, Sα, Ka}. They correspond to the
translation Pa, the Lorentz rotation Mab, the SUSY Qα, the dilatation D, the chiral U(1)
A, the SSUSY Sα and the conformal boost Ka, respectively. Such additional gauge degrees
of freedom are technically useful for the construction of the SUGRA action. In conformal
SUGRA, a supermultiplet is characterized by the charges under D and A denoted by w and
n, respectively. We introduce one particular supermultiplet called the compensator, whose
components are auxiliary fields or removed by the superconformal gauge fixing. In this
paper, we use a chiral superfield as the compensator, which gives the socalled oldminimal
SUGRA after the superconformal gauge fixing.
In the following, we summarize the component expressions of supermultiplets, the
chiral projection operation, the invariant formulae and some identities.
General multiplet. The components of a general multiplet with charges (w, n) are given
by
whose charge conjugate C¯ is
C = (C, ζ, H, K, Ba, λ, D) ∈ G(w,n),
C¯ = (C∗, ζc, H∗, K∗, Ba∗, λc, D ) ∈ G(w,−n),
∗
where ζc and λc are the charge conjugates of ζ and λ, respectively. Note that C¯ has the D
and A charges (w, −n).
Multiplication law.
Here We show the multiplication rule of supermultiplets. Suppose
CI ∈ G(wI ,nI ) and consider a function f (CI ) ∈ G(w,n). The component of f (CI ) is given by
f (CI ) = f (CI ) , fI ζI , fI HI + · · · , fI KI + · · · , fI BaI + · · · , fI λI + · · · ,
fI DI +
1
2 fIJ HI HJ + KI KJ
− BaI BJa
− DaCI DaCJ + · · · ,
where ellipses denote terms containing fermions and fI , fIJ are derivatives defined as
and the covariant derivative of CI is given by
fI ≡
∂f (C)
∂CI ,
fIJ ≡ ∂CI ∂CJ
,
∂2f (C)
DaC = eaμ(∂μ − ωbμ − inAμ)C + · · · .
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
Note that since (w, n) are additive quantum numbers, the following relations are satisfied,
X wI fI C
I = wf (C),
X nI fI C
I = nf (C),
X wJ fIJ CJ = (w − wI )fI (C). (A.6)
I
I
J
Chiral multiplet and chiral projection. The components of a chiral superfield are
satisfies the constraint w = n(−n) and can be embedded into a general multiplet (A.1) as
√2iPLχ , −F , iF , iDaφ , 0 , 0
∈ G(w,w),
whereas an antichiral multiplet Φ¯ = (φ∗, PRχ , F ∗) ∈ Σ¯ w can be embedded as
φ¯ , √2iPRχ , −F¯ , −iF¯ , −iDaφ¯ , 0 , 0
∈ G(w,−w).
One can make a chiral multiplet out of a general multiplet satisfying w − n = 2, and
we refer to this operation as the chiral projection Σ
Σ : C ∈ G(w,w−2) →
Σ(C) ∈ Σw+1,
(H − iK) , √ PL(λ + γaDaζ) , − 2 (D + DaDaC + iDaBa) .
2
1
whose components are given by
Σ(C) =
1
2
where
In particular, for C ∈ G(2,0), we find that
DaDaC + iDaBa + c.c. = − 3
where we have used
DaDaC = eμa (∂μ − (w + 1)bμ − inAμ) DaC − ωaabDbC + 2wfaaC + · · · , (A.12)
DaBa = eaμ(∂μ − (w + 1)bμ − inAμ)Ba − ωaabBb + 2infaaC + · · · .
Here the ellipses denote terms containing fermions, which we do not focus on in this paper.
R(C + C¯) + e−1∂μ (eeμa(DaC + iBa + c.c.)) + · · · , (A.14)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
(A.13)
(A.15)
(A.16)
(A.17)
i
1
Σ(CΦ) = Σ(C)Φ ∈ Σn+m+1.
faa = − 12
1
R + · · · .
For instance, we can construct a chiral superfield out of a chiral and an antichiral superfield:
Φ ∈ Σ0,
S0 ∈ Σ1,
→
′
Φ =
Σ(S¯0Φ¯ )
S2
0
∈ Σ0.
Note that the chiral projection does not act on a chiral multiplet, i.e. for C ∈ Gn,n−2, Φ ∈
Σm, we find that
Vector multiplet V ∈ G(0,0) and gauge transformation.
We define a gauge vector
V ∈ V :
V ∈ G(0,0),
V = V¯
The composite supermultiplet Φ¯ e2gV Φ is invariant under the SUSY gauge transformation
e2gV
→ e2gV ′ = e−gΛ¯ e2gV e−gΛ,
′
Φ → Φ = egΛΦ,
Φ¯ → Φ¯′ = Φ¯ egΛ¯ .
Under this transformation, a chiral supermultiplet Φ˜ ≡ Σ(Φ¯ e2gV ) transforms as
˜
Φ
→
Φ˜ ′ = Σ Φ¯ e2gV e−gΛ
= Φ˜ e−gΛ.
We take the WessZumino gauge, in which the components of V are given by
V WZ = [Bμg , λg, Dg],
C(V WZ) = [0, 0, 0, 0, Bag, λg, Dg].
Here the ordinary gauge transformation with Λ = [iθ, 0, 0] is given by
Invariant action formula. The superconformal invariant actions are given by the
Dand F term density formulas: the Dterm invariant formula with C ∈ G(2,0) and C¯ = C is
[Σ(C)]F = − 2 [C + C¯]D
1
for C ∈ G(2,0).
Σ(X¯ )T F = Σ(X¯ T ) F = − 2
X¯ T + T¯X D
.
1
h
Σ Φ¯ e2gV Φ˜ i
F
= − 2
1 hΦ¯ e2gV Φ˜ + c.c.i .
D
(A.18)
(A.19)
(A.20)
(A.21)
(A.22)
(A.23)
(A.24)
(A.25)
(A.26)
(A.27)
and the F term invariant formula with Φ ∈ Σ3 is
There is a useful identity between these two invariant formulas
Then, for T, X ∈ Σ1,
In addition, for U(1) charged chiral multiplets Φ, Φ˜ ∈ Σ1, we find that
Composite supermultiplets.
We finally show the components of composite superfields.
With Φ¯ ∈ Σ¯ 1, we can make a chiral superfield with w = n = 2 as
Σ(Φ¯ ) =
−F¯ , PLγaDaχ , −DaDaφ¯ ∈ Σ2.
The composite antichiral superfield Φ¯ e2cV
∈ Σ¯ 2 can be embedded into a general multiplet
C(Φ¯ e2cV ) WZ = hφ¯ , √
2iPRχ , −F¯ , −iF¯ , −iDaφ¯ + 2cBagφ¯ + · · · ,
Note that c denotes a gauge charge of Φ under V . In addition, the projected composite
superfield takes the form
· · · , 2cDgφ¯ + 2icBagDaφ¯ − 2φ¯c2BagBgai.
(A.29)
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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