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PHYSICAL REVIEW D 101, 075012 (2020)

Minimal Starobinsky coupled to a - superfield

† ‡ Yermek Aldabergenov ,1,2,* Shuntaro Aoki ,3, and Sergey V. Ketov 4,5,6, 1Department of Physics, Faculty of Science, Chulalongkorn University, Thanon Phayathai, Pathumwan, Bangkok 10330, Thailand 2Institute for Experimental and , Al-Farabi Kazakh National University, 71 Al-Farabi Avenue, Almaty 050040, Kazakhstan 3Department of Physics, Waseda University, Tokyo 169-8555, Japan 4Department of Physics, Tokyo Metropolitan University, 1-1 Minami-ohsawa, Hachioji-shi, Tokyo 192-0397, Japan 5Research School of High-Energy Physics, Tomsk Polytechnic University, 2a Lenin Avenue, Tomsk 634028, Russian Federation 6Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, Kashiwa 277-8583, Japan

(Received 2 February 2020; accepted 13 March 2020; published 8 April 2020)

The minimal Starobinsky supergravity with (scalaron) and goldstino in a massive vector supermultiplet is coupled to the dilaton-axion chiral superfield with the no-scale Kähler potential and a . The Kachru-Kallosh-Linde-Trivedi-type superpotential with a constant term is used to stabilize dilaton and axion during , but it is shown to lead to an instability. The instability is cured by adding the alternative Fayet-Iliopoulos (FI) term that does not lead to the gauged R . Other stabilization mechanisms, based on the Wess-Zumino-type superpotential, are also studied in the presence of the FI term. A possible connection to the D3- models is briefly discussed too.

DOI: 10.1103/PhysRevD.101.075012

I. INTRODUCTION implies the necessity to extend viable infla- tionary models to N ¼ 1 supergravity in four spacetime Cosmological inflation offers a simple solution to basic as the first step. A supergravity extension of the problems of standard cosmology and current cosmological Starobinsky inflationary model is not unique, being de- observations of the cosmic microwave background (CMB) pendent upon the supergravity framework chosen; see, e.g., radiation. Viable theory models of inflation are Ref. [5] for a review. usually nonrenormalizable after quantization, which raises The minimal description of Starobinsky inflation in a problem of their ultraviolet (UV) completion in quantum supergravity as a single-field inflationary model (with a . It is important because physical predictions single physical scalar called scalaron) is possible when the of those models are known to be sensitive to quantum scalaron (inflaton) is assigned to a massive Abelian vector corrections. Amongst all viable inflationary models, multiplet [6–9] in terms of unconstrained superfields. Starobinsky’s model [1] attracted a lot of attention because A generic action of a vector multiplet V is governed by it provides (so far) the best fit to the cosmological a single (real) potential JðVÞ, while its bosonic part in observations [2]. The Starobinsky inflationary model of Einstein frame reads (M ¼ 1)1 modified (R þ R2) gravity and its scalar-tensor gravity P counterpart are nonrenormalizable indeed, with the UV 1 1 1 cutoff being Planck mass M [3,4]. Assuming quantum −1L − mn − ∂ ∂m P e bos ¼ 2 R 4 FmnF 4 JCC mC C gravity to be given by , the UV completion in 1 − m − 2 4 JCCBmB 8 JC; ð1Þ *[email protected] † [email protected] ‡ [email protected] where R is Ricci scalar, C is the leading field component of V, Bm is an Abelian vector field with the Abelian Published by the American Physical Society under the terms of field strength Fmn ¼ ∂mBn − ∂mBn and the gauge the Creative Commons Attribution 4.0 International license. g, and the subscripts (C)ofJ denote Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 1We use the spacetime signature ð−; þ; þ; þÞ.

2470-0010=2020=101(7)=075012(11) 075012-1 Published by the American Physical Society ALDABERGENOV, AOKI, and KETOV PHYS. REV. D 101, 075012 (2020) the derivatives of J with respect to C. The scalar potential in KðΦ; Φ¯ Þ¼−n log ðΦ þ Φ¯ Þ; ð3Þ Eq. (1) is obtained after elimination of the auxiliary field D of the vector multiplet, so that is of the D type. The famous where we have introduced the real parameter n>0.In Starobinsky inflationary scalar potential is obtained by the context of string theory, the no-scale Kähler choosing the J potential as potential arises in toroidal (and ) compactifications pffiffiffiffiffiffiffiffi  of type II strings and in the large volume limit of Calabi- JðCÞ¼−3ðC þ lnð−CÞÞ with C ¼ − exp 2=3φ Yau compactifications of heterotic strings (specifically, with n ¼ 1 for dilaton-axion and with n ¼ 3 for a volume ð2Þ modulus). The superpotential of dilaton and axion in string theory may only be generated nonperturbatively. It is in terms of the canonical scalaron φ. Then the first term in common in the literature to assume its specific form either JðCÞ is responsible for the induced as a Wess-Zumino-type cubic polynomial or as an expo- driving inflation, whereas the second term in JðCÞ repre- nential of Φ. sents an exponentially small correction during slow roll of At first sight, adding those couplings is not a problem the scalaron for positive values of φ, which is responsible in supergravity. However, we find that it spoils the for a suppression of the tensor-to-scalar ratio in the Starobinsky inflation because of an instability. This phe- Starobinsky inflation; see e.g., Ref. [10] for details. nomenon was first observed in Ref. [26] in the context of Since inflation is driven by positive energy, in super- the so-called Polonyi-Starobinsky supergravity where a gravity, it leads to a spontaneous breaking Polonyi chiral superfield with the canonical kinetic term (SSB). Therefore, a Goldstino should be present during and a linear superpotential were introduced for describing inflation. In the minimal supergravity description of infla- SSB and dark after inflation [27–29] toward tion, the Goldstino is given by (“”) that is combining our (early time) inflationary models with late the of scalaron. It is worth to recall that the time cosmology. In the case of the no-scale Kähler Goldstino action is universal, being given by the Akulov- potential, we find a different situation because both dilaton Volkov action [11] up to a field redefinition [12,13]. and axion have to be trapped near a minimum of their scalar Since an Abelian vector multiplet is always present in potential during the Starobinsky inflation driven by the the world volume of the spacetime-filling D3 brane (or anti- scalaron, i.e., the masses of both dilaton and axion have to D3 brane) [14–16], the D3-brane effective action may be larger than the Hubble scale during inflation (it is known provide the desired embedding of the Starobinsky infla- as the stabilization in the literature [30]). It is the tionary model into string theory. This conjecture is strongly purpose of this paper to achieve the moduli stabilization of supported by the existence of the Bagger-Galperin (BG) dilaton and axion with the Kähler potential (3) by using a action [17] of an N ¼ 1 Abelian vector mulitplet, which is suitable superpotential and the alternative FI term in the the Dirac-Born-Infeld-type (DBI-type) extension of the minimal Starobinsky supergravity coupled to the dilaton- standard N ¼ 1 Maxwell-type action, because the BG axion superfield.2 Our setup and motivation are different action has the second (nonlinearly realized) supersymmetry from those of Ref. [25] where inflaton is identified with needed for its D-brane interpretation. However, in order to dilaton. They are also different from those of Ref. [31], prove the conjecture, one needs to (i) realize Starobinsky’s where the superpotential is chosen in Polonyi’s form. inflation in the DBI framework, (ii) provide SSB of the Though we did our calculations with the DBI kinetic first (linearly realized) supersymmetry, and (iii) restore the terms for the vector multiplet, in this paper we only use the second (nonlinearly realized) supersymmetry after cou- Maxwell-type kinetic terms for simplicity, because the DBI pling the BG (or DBI) action to supergravity. structure does not significantly affect the Starobinsky The first problem was already solved in Refs. [18,19]. The inflation and the moduli stabilization in question, according viable SSB after the Starobinsky inflation, which gives rise to our findings (see also Refs. [18,23,28] for more). to an adjustable (or observable) cosmological constant, is The paper is organized as follows. In Sec. II, we recall also possible by the use of the alternative Fayet-lliopoulos the superconformal tensor calculus in N ¼ 1 supergravity (FI) terms without the gauging of the R symmetry [20–23]. and introduce our notation. In Sec. III, we define our However, an origin of those FI terms in string theory and a inflationary model of the minimal Starobinsky supergrav- restoration of another supersymmetry are still unclear; see, ity, whose vector (inflaton) multiplet is coupled to the e.g., Refs. [24,25] for recent developments. dilaton-axion chiral superfield with the no-scale Kähler In this paper, we do not address those unsolved pro- potential. The vacuum structure of the model with the blems but check whether the minimal formulation of the Kachru-Kallosh-Linde-Trivedi (KKLT)-type superpotential Starobinsky inflation in supergravity is compatible with its Φ coupling to the chiral dilaton-axion superfield , described 2We call Φ to be the dilaton-axion superfield for simplicity, by the no-scale Kähler potential K and a superpotential W. although it may also represent a moduli superfield, with a generic The no-scale Kähler potential reads parameter n>0.

075012-2 MINIMAL STAROBINSKY SUPERGRAVITY COUPLED … PHYS. REV. D 101, 075012 (2020) toward stabilization of dilaton and axion is investigated in where the index i ¼ 1; 2; 3; …, counts the matter multip- Sec. IV, where an instability of inflation is found. A cure to lets. The S0 has the weights (1,1) and is used to fix some of the instability is proposed in Sec. V by using the alternative the superconformal . The matter multiplets Si FI term. Section VI is our conclusion. In the Appendix A, have the weights (0,0). The antichiral multiplets are ¯ ¯ i¯ we study a different stabilization mechanism by using the denoted by S0 and S . Wess-Zumino (WZ)-type superpotential. Another stabili- As regards a general (real) multiplet, it has the field zation mechanism, as a combination of the previous ones, content is proposed in Appendix B. Spontaneous after inflation is studied in Appendix C. V ¼fC; Z; H; K; Ba; Λ; Dg; ð5Þ II. SUPERCONFORMAL TENSOR CALCULUS where Z and Λ are , Ba is a (real) vector, and The conformal N ¼ 1 supergravity techniques are others are (real) scalars, respectively. described in Refs. [32–36]. We follow the notation and The (gauge) field strength multiplet W has the weights conventions of Ref. [37]. In addition to the local symmetries (3=2, 3=2) and the following field components: of Poincar´e supergravity, one also has the gauge invariance     1 1 under dilatations, conformal boosts, and S supersymmetry, as η¯W η¯P λ;pffiffiffi − γ Fˆ ab iD P η;η¯P Dλ ; 6 1 ¼ L 2 ab þ L L ð Þ well as under Uð ÞA rotations. The gauge fields of dilatations 2 and Uð1Þ rotations are denoted by bμ and Aμ, respectively. A ˆ A multiplet of conformal supergravity has charges with where η is the dummy , Fab ¼ ∂aBb − ∂bBa þ 1 ψ¯ γ λ ≡ ψ¯ γ λ respect to dilatations and Uð ÞA rotations, called Weyl and ½a b Fab þ ½a b is the superconformally covariant chiral weights, respectively, which are denoted by pairs field strength, ψ a is , λ and D are Majorana (Weyl weight, chiral weight) in what follows. and the real auxiliary scalar, respectively. The A chiral multiplet has the field components, related expressions of the multiplets W2 and W2W¯ 2, which are embedded into the chiral multiplet (4) and the general χ S ¼fS; PL ;Fg; ð4Þ multiplet (5), respectively, are   where S and F are complex scalars, and PLχ is a left- 1 W2 ; ; FF − FF˜ − D2 ; 7 handed Weyl fermion (PL is the chiral projection ). ¼ þ2 ð Þ ð Þ In this paper, we use the two types of chiral multiplets: the i conformal compensator S0 and the matter multiplets S ,

  1 2 ¯ 2 − ˜ − 2 2 2 W W ¼ ; ; ; ; ; ; þ2 jðFF FFÞ D j ; ð8Þ where we have omitted the fermionic terms (denoted by dots) for simplicity. In addition, we use the book-keeping notation ab ˜ ab i abcd FF ¼ FabF and F ≡ − 2 ε Fcd throughout the paper. We also need another chiral multiplet,

  1 FF 1 FF˜ − D2 Σ ¯ 2 4 − ð2 þ 2 Þ F0 ˜ − 2 2 ðW =jS0j Þ¼ 4 þ; ; 4 ðFF þ FF D Þþ ; ð9Þ jS0j jS0j S0 where Σ is the chiral projection operator [35,36]. The of the weights (2,0). Here, the dots in the higher compo- argument of Σ requires specific Weyl and chiral weights: in nents also include some bosonic terms, but we do not write order to make sense to ΣV, the V must satisfy w − n ¼ 2, down them here for simplicity (see Ref. [20] for their where ðw; nÞ are the Weyl and chiral weights of V.We explicit expressions). adjust the correct weights of the argument by inserting the A massive vector multiplet V has the field components, 4 factor jS0j . Equation (9) is the conformal supergravity counterpart of the superfield D¯ 2W¯ 2. V C; Z; H; K; B ; λ;D ; 11 The of W is given by [36] ¼f a g ð Þ

while all of them are either real (bosonic) or Majorana D −2 W ¼f D; ; ; ; ; ; g ð10Þ (fermionic). The weights of V are (0,0).

075012-3 ALDABERGENOV, AOKI, and KETOV PHYS. REV. D 101, 075012 (2020)

The bosonic parts of the F-term invariant action The subscripts of J denote the derivatives of J with formulas are respect to the scalar fields C, Φ, and Φ¯ , respectively. The V Z is the scalar potential, whose explicit form reads 1 ffiffiffiffiffiffi 4 p− ¯ ½SF ¼ d x gðF þ FÞ; ð12Þ 2 V ¼ VF þ VD; ð18Þ     while they can be applied only when S has the weights J 2 J J −1 W J W 2 C − 3 W 2 (3,3). The bosonic part of the D-term formula for a real VF ¼ e ð ΦΦ¯ Þ j Φ þ Φ j þ j j ; J CC multiplet ϕ of the weights (2,0) is ð19Þ Z   ffiffiffiffiffiffi 1 ϕ 4 p− − ω 2 ½ D ¼ d x g Dϕ CϕRð Þ ; ð13Þ J 3 V ¼ C ; ð20Þ D 8ðΦ þ Φ¯ Þ where RðωÞ is the superconformal Ricci scalar in terms of in agreement with Refs. [8,9,31]. When W 0, the spacetime metric and bμ [37]. The Cϕ and Dϕ are the first ¼ and the last components of ϕ, respectively. Lagrangian (17) reduces to the one of Ref. [38] considered in the context of global supersymmetry. In the absence of We set the (reduced) MP and the Abelian Φ gauge coupling constant g to unity for simplicity in our , the equations above reduce to Eqs. (1) and (3). The Planck mass M can be recovered as follows: calculations, unless it is stated otherwise. Both of them can P be restored by dimensional considerations and rescaling of ffiffi ffiffi rffiffiffi p2 φ Φ p2 ϕ 2 C 3 − a the vector multiplet fields, respectively. ¼ −e MP ; ¼ e nMP þ i : ð21Þ MP MP n MP III. THE MODEL The fields φ; ϕ, and a can be identified as the canonical Let us consider the supergravity model of a massive inflaton/scalaron, dilaton, and axion, respectively. After vector multiplet V coupled to a dilaton-axion chiral rewriting the Lagrangian in terms of those fields, we obtain Φ ffiffi multiplet , whose action is given by p ϕ 1 2 1 2 1 2 2 2 L ¼ − ð∂ φÞ − ð∂ ϕÞ − e nMP ð∂ aÞ − V − V ; 3 2 a 2 a 2 a F D − 2 −J =3 2 3W − Φ 2 S ¼ 2 ½jS0j e D þ ½S0 F ½ W F; ð14Þ ð22Þ where J is a real function of C and (Φ þ Φ¯ ). We take J as where (after a restoration of the gauge coupling constant g a sum of the Starobinsky potential (2) and the no-scale also) we have Kähler potential (3), pffiffi  −2 2 ϕ ffiffi   nM p ϕ ¯ J 2 e P 2 − 2 W W C ¯ =M W − nM W W¯ J −3 −Ce − n Φ Φ ; VF ¼ e P j Φj e P Φ þ Φ¯ ¼ log ð Þ logð þ Þ ð15Þ n M M  P P pffiffi pffiffi 2 2 φ 2 2 φ jWj where n is a positive integer. The first term in Eq. (15) is −6 3M 3 3M þðn e P þ e P Þ 2 ; ð23Þ supposed to be responsible for the Starobinsky-type infla- MP tion, and the second one describes the interactions of 2 4 pffiffi pffiffi  W 9 2 ϕ 2 φ 2 dilaton and axion. The is a holomorphic superpotential g MP − 3 V ¼ e nMP 1 − e MP : ð24Þ depending on Φ only. D 16 Our action is invariant under a constant shift The Starobinsky inflation is supposed to be driven by the Φ → Φ þ ic ð16Þ D-type term above. However, in the case under consider- ation, the D term has the dilaton-dependent factor. Therefore, with a real constant c, except the superpotential term. a viable inflation is only possible after a stabilization of the After imposing the superconformal gauge fixing and dilaton, while keeping the F term to be relatively small integrating out the auxiliary fields, the bosonic part of the against the D term, so that the F term should not spoil the action (14) is given by Starobinsky inflation either.

1 1 1 IV. THE VACUUM STRUCTURE 2 2 a ¯ L ¼ R − J CCð∂aCÞ − J CCBa − J ΦΦ¯ ∂aΦ∂ Φ − V 2 4 4 We choose the KKLT-type superpotential [39] as our 1 1 − ðΦ þ Φ¯ ÞFF þ ðΦ − Φ¯ ÞFF:˜ ð17Þ ansatz to safely stabilize dilaton and axion in our model 4 4 (Sec. III).

075012-4 MINIMAL STAROBINSKY SUPERGRAVITY COUPLED … PHYS. REV. D 101, 075012 (2020)  Let us first study the stationary conditions of all fields. 1 4A2B2ρ2 pffiffi J −2Bρ 2φ VF ¼ e n e In terms of the inflaton C ¼ −e 3 , the dilaton ReΦ ¼ ρ, ð2ρÞ n 3 Φ θ −2Bρ Bρ and the axion Im ¼ , the scalar potential is given by þ 4ABρe ðA þ W0e cosðBθÞÞ  2 −Bρ 2 −2Bρ 2 pffiffi  P n W 2AW0e cos Bθ A e : 9g − 2φ 2 þð þ Þð 0 þ ð Þþ Þ 1 − 3 VD ¼ 16ρ e ; ð25Þ ð31Þ

 θ 2 π ∈ Z 1 2ρ 2 We find that ¼ m ;m , minimizes the potential for J ð Þ 2 ¯ ¯ V ¼ e jWΦj − 2ρðWWΦ¯ þ WWΦÞ n ≥ 3 since P þ n ≥ 0 holds in that case. In the following, F ð2ρÞn n  we focus on the point at θ ¼ 0. Taking into account the condition φ ¼ 0 for Vφ ¼ 0, the condition Vρ ¼ 0 is þðP þ nÞjWj2 ; ð26Þ reduced to either of

Bρ Að2Bρ þ nÞþnW0e ¼ 0; ð32Þ where we have introduced the notation 4 ρ − 3 4 ρ ρ − 1 2 − 3 Bρ 0 pffiffi ffiffiffi Aðnð B Þþ B ðB Þþn Þþðn ÞnW0e ¼ : 2φ p φ 3 3 − 6φ Jð Þ¼ e ; ð27Þ ð33Þ

pffiffi pffiffi  These conditions should be regarded as the equations that 2φ 2φ PðφÞ¼3e 3 e 3 − 2 ð28Þ determine the of ρð¼ ρ0Þ. In what follows, we consider the no-scale case with n ¼ 3 for definiteness. Then, Eq. (32) becomes and have recovered the gauge coupling constant g that   2 determines the inflationary scale. −Bρ0 W0 −Ae 1 Bρ0 ; The first derivative of the scalar potential with respect to ¼ þ 3 ð34Þ φ reads that is exactly same as the KKLT vacuum.5 The relevant rffiffiffi 2 pffiffi  masses at the stationary point are explicitly given by 9g 2 2 1 − 3φ J 2 Vφ ¼ φ 1 − e þ JφVF þ e PφjWj ; 8ρ 3 2ρ n 2 2 2 3−2Bρ0 ð Þ 2 27g − 4A B e mφ ¼ ; ð29Þ 36ρ0

2 2 3−2Bρ0 2 A B e ðBρ0 þ 2Þð2Bρ0 þ 1Þ mρ ¼ 3 ; where the subscripts denote the derivatives with respect to a 6ρ0 given field. The φ ¼ 0 is a solution of Vφ ¼ 0, since Jφ and 2 3 3−2Bρ0 A B e ð2Bρ0 þ 3Þ φ 0 2 Pφ vanish at ¼ . mθ ¼ 6ρ2 ; ð35Þ Let us assume that the superpotential takes the following 0 form: and they are all positive.6 The minimum is of the anti–de Sitter (AdS)-type because the cosmological constant is −BΦ W ¼ W0 þ Ae ð30Þ given by

A2B2e3−2Bρ0 that is inspired by the KKLT-type superpotential [39] with V0 ¼ − < 0; ð36Þ 4 6ρ0 constant parameters W0, A, and B. The nonvanishing constant W0 is essential in our investigation. We assume while supersymmetry is restored at the minimum. that W0 is negative and A; B are both positive. In this case, Though both dilaton and axion are stabilized by using the F-term scalar potential is explicitly given by the KKLT-type superpotential, as was demonstrated above, there is still a problem. In the Starobinsky-type inflationary 3 V ≫ Canonical normalizations of the dilatonpffiffi and axionqffiffi fields are scenario, it is necessary to require D jVFj. But the − 2ϕ 2 Φ n obtained after a field redefinition ¼ e þ i na. We find ρ θ 5 ρ 0 − 2 convenient to use and in what follows. Equation (33) yields the solutions ¼ ; B. 4Another type of the superpotential is considered in 6We assume that the inflationary scale ∼g is larger than that Appendix A. of VF.

075012-5 ALDABERGENOV, AOKI, and KETOV PHYS. REV. D 101, 075012 (2020) double exponential in the eJ factor and the exponentials in Let us choose J and ξ so they satisfy the relation the P function, defined by Eqs. (27) and (28) in terms of the   canonical inflaton φ, destroy the flatness of the scalar ξðCÞ 1 J þ ¼ −3 1 þ ; ð38Þ potential and thus greatly reduce the e-foldings number of C 3g C inflation. Therefore, we need the hierarchy of the two parameters, namely, g ≫ A. However, as can be seen from where we have set Eq. (35), it gives rise to the extremely small dilaton and   1 axion masses and, therefore, the KKLT stabilization ansatz kJ ξðCÞ¼3gξ0e 1 þ ; ξ0 < 0;k>0: ð39Þ alone does not work here.7 C

The case ξðCÞ ∝ ekJ was studied in Ref. [26]. In Eq. (39), V. THE FIELD-DEPENDENT FI TERM 1 1 ξ −1 0 we added the factor ð þ CÞ to ensure ð Þ¼ . This As long as the total J potential (2) is unchanged, it gives factor does not change the results of Ref. [26] since it is rise to the Starobinsky D-type inflationary potential, as reduced to 1 for C → −∞. However, due to a change of the desired. However, as we found in the previous section, the J function in Eq. (38), the canonical scalaron field in coupling of the vector (inflaton) superfield to the chiral Eq. (21) has to be modified. We find convenient to use the (dilaton-axion) superfield converts a single-field inflation noncanonical inflaton field C in what follows. into a multifield inflation, while it leads to the instability For large negative C, Eq. (38) can be approximately resulting in a significant reduction of the duration of solved as inflation, measured by the e-foldings number Ne, and, 1 1 hence, to an unacceptable change in the predicted CMB 3kðC−C0Þ J ∼− log ðe − ξ0Þ; ð40Þ power spectrum measured by the scalar index ns and the k 3 tensor-to-scalar ratio r, both depending upon Ne. Our idea to solve this problem is to change the origin of the first term with the integration constant C0. Thus, the function J in the J potential (2) and thus avoid the instability via the becomes constant as induced change in the F-type scalar potential. 1 −ξ0 Let us introduce the following alternative field- J∞ ≡ − log ; ð41Þ dependent FI term (cf. Refs. [20–22]):8 k 3 for C → −∞. Then the exponential factor eJ in the F-type   3 2 ¯ 2 part VF of the scalar potential (26) also becomes constant 2 −J =3 W W 2 S ¼ − jS0j e ξ DW ; ð37Þ − 3 FI 2 D 2 D¯ ¯ 2 during inflation, while the term PðCÞ¼JC=JCC in the ð WÞ ð WÞ D VF becomes constant too,

kJ∞ ξ 3 − 2ξ0e where is a real function that, in general, depends on C P∞ ¼ : ð42Þ ξ kJ∞ and a combination (Φ þ Φ¯ ), in order to preserve the k 0e shift symmetry in Eq. (16). This FI term does not require the gauged R symmetry, and therefore, is applicable To summarize, we obtain the following full scalar together with our KKLT-type superpotential.9 It appears potential during inflation: that a constant ξ does not help because it merely shifts the    9g2 1 2 1 2ρ 2 J J∞ ð Þ 2 ¯ vacuum and does not contribute to e and P.Aswas V ¼ 1 þ þ e jWΦj − 2ρðWWΦ¯ 16ρ C ð2ρÞn n noticed in Ref. [26], the dangerous terms in the scalar  ξ potential can be removed when and J satisfy a specific ¯ 2 relation, by extending ξ to be field dependent. Here we þ WWΦÞþðP∞ þ nÞjWj ; ð43Þ apply the same idea to the case under consideration, where the dilaton-axion multiplet is coupled to the massive vector whose F term in the second line does not spoil the multiplet. Starobinsky inflation described by the D term in the first J∞ line because e and P∞ are constants during the inflation. 7 φ The observational predictions for the cosmological tilts n The range of the scalaron field =MP is trans-Planckian of the s order Oð1Þ during the (large field) Starobinsky inflation [19]. and r with the e-folding number Ne between 50 and 60 in 8The standard FI term [40] in the context of supergravity this inflationary model are the same as in the Starobinsky does not work because it implies the gauged R symmetry and, case; see Ref. [26] for details. hence, the charged gravitino field that severely restricts possible couplings. As the inflation ends, the inflaton C and the dilaton- 9The gauged R symmetry does not allow a constant term in the axion Φ ¼ ρ þ iθ take the vacuum expectation values superpotential. which are determined by the vacuum conditions

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1 ffiffiffiffiffiffi VC ¼ Vρ ¼ Vθ ¼ 0. We find that C ¼ −1 is still a solution p μν − −gFμνF to V ¼ 0 since J j −1 ¼ P j −1 ¼ 0 in the paramet- 4 C C C¼ C C¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rization of Eq. (39). As regards Vρ ¼ Vθ ¼ 0, the results of → 4 − − − −2 0 MBI detðgμνÞ detðgμν þ MBI FμνÞ ð47Þ the previous Sec. IV apply since VDjC¼−1 ¼ . Moreover, we do not have to demand VD ≫ jVFj with the FI term because the structure of J and ξ already solves 4 with the dimensional deformation parameter MBI, is avail- the problem. Therefore, we can strongly stabilize Φ by able and does not significantly change our results. The DBI choosing the superpotential parameters appropriately, i.e., structure is, however, relevant for a possible embedding of with a sufficiently large A in Eq. (30). our model into the effective action of a D3 brane. For completeness, we provide the masses of all scalar The BI action is known to have the Uð1Þ electric- fields in the vacuum. The masses given in Eq. (35) get small magnetic self-duality, while its minimal coupling to the modifications due to the change of the J function; see massless dilaton and axion results in the SLð2; RÞ self- Eq. (38). They are explicitly given by duality [41] that also applies to the D3-brane effective   action of the massless fields. The manifestly N ¼ 1 locally 9g2 A2B2eJð−1Þ−2Bρ0 1 supersymmetric extension of the BI action, coupled to the 2 − 1 ξ kJð−1Þ mC ¼ þ 0e ; 8ρ0 6ρ0 3 massless dilaton-axion chiral superfield and preserving the SLð2; RÞ self-duality, can be found in Ref. [42].10 The self- 2 2 Jð−1Þ−2Bρ0 2 A B e ðBρ0 þ 2Þð2Bρ0 þ 1Þ mρ ¼ ; duality properties are only valid in the case of the massless 6ρ3 11 0 fields and in the absence of a superpotential. 2 3 Jð−1Þ−2Bρ0 A B e ð2Bρ0 þ 3Þ Another (nonlinearly realized) supersymmetry is also m2 ; θ ¼ 6ρ2 ð44Þ required for a D3 brane. Our supergravity model has 0 manifest N ¼ 1 supersymmetry but does not have another supersymmetry by construction, though it may still be where Jð−1Þ is the value at C ¼ −1. In the limit ξ0 → 0, possible after a modification of our action or by using these masses are reduced to those in Eq. (35). nonlinear realizations where manifest supersymmetry is A comment is in order here. Consistency of the alter- absent. Unlike the standard FI term, the alternative FI terms native FI term (37) requires the vacuum expectation value avoid the no-go theorems known in supergravity and string of the D term to be nontrivial; in other words, supersym- theory [44] so that the search for an origin of the alternative metry must be spontaneously broken in the vacuum. Since FI term (37) in string theory deserves further investigation. D in our case is given by Finally, we mention a possible connection to extended   supersymmetry and supergravity. Some alternative FI terms g ξðCÞ 2 2 D ¼ JC þ ; ð45Þ were recently found in N ¼ supergravity [45]. The N ¼ 2ρ 3g supersymmetric extensions of the BI theory both in super- space and via nonlinear realizations also exist [46–50]. The the vacuum discussed above is not allowed. To be con- scalar (ϕiÞ kinetic terms of the N-extended vector multiplet sistent, the right-hand side of Eq. (38) should be modified enter the generalized BI action via the root further as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   4 −2 i i −M − det ½gμν þ M ðFμν þ ∂μϕ ∂νϕ Þ; ð48Þ ξðCÞ 1 BI BI J þ ¼ −3 1 þ þ Δ; ð46Þ C 3g C which is different from the k inflation [51] and Horndeski gravity [52]. where a nonvanishing (small) constant Δ has been intro- duced. As long as jCΔj ≪ 1 during inflation, the proposed mechanism applies without changing the results above. The ACKNOWLEDGMENTS detailed analysis is given in Appendix C. The authors are grateful to Hiroyuki Abe for his collabo- ration on related work in Refs. [23,28] and Sergei Kuzenko VI. CONCLUSION for correspondence. S. A. was supported in part by a Waseda University Grant for Special Research Projects (Project In this paper, we studied the phenomenological aspects No. 2019E-059). Y.A. was supported by the CUniverse of inflation in our new supergravity model. Our main research promotion project of Chulalongkorn University in results are summarized in the Abstract. As we already mentioned in the Introduction, the DBI 10See also the related construction in Ref. [43] toward the deformation of the vector multiplet kinetic terms in our electromagnetic confinement. supergravity model, which is essentially described by a 11Both dilaton and axion are massless in the perturbative string locally supersymmetric extension of the BI action theory.

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Bangkok, Thailand, under the grant reference CUAASC, and The vacuum is AdS, whose depth is given by by the Ministry of Education and Science of the Republic of 1 Kazakhstan under the Grant No. BR05236322. S. V.K. was − 3 2 V0 ¼ 4 e m : ðA8Þ supported by Tokyo Metropolitan University, the World Premier International Research Center Initiative, MEXT, The situation can be improved by adding quartic Japan, and the Competitiveness Enhancement Program of couplings inside the log of the Kähler potential in the J Tomsk Polytechnic University in Russia. function. Let us modify J as ¯ ¯ 4 J ¼ JðCÞ − n log½Φ þ Φ þ γ1ðΦ þ Φ − 2ρ0Þ APPENDIX A: THE WZ-TYPE ¯ 4 þ γ2ðΦ − Φ − 2iρ0Þ ; ðA9Þ SUPERPOTENTIAL γ Let us investigate another case of the Wess-Zumino where 1;2 are the real parameters. After these modifica- (WZ)-type superpotential in order to stabilize dilaton and tions, the stationary point is the same as in the model axion during Starobinsky inflation in our supergravity without the quartic modifications, i.e., model. We did our calculations with a generic (cubic) φ ¼ 0; ρ ¼ θ ¼ ρ0; ðwith n ¼ 4Þ: ðA10Þ WZ superpotential, but those results are cumbersome and not very illuminating. We restrict ourselves in this The quartic couplings affect the mass terms that can be Appendix to the most relevant mass term for simplicity, i.e., roughly evaluated as

2 Δ 2 ∼ J −1 2 W ¼ mΦ ; ðA1Þ m ð ΦΦ¯ ÞΦΦ¯ jFΦj : ðA11Þ Thus, the messes get no corrections when the vacuum where m is a real constant. The F-term potential becomes preserves supersymmetry, like in the KKLT case. In the   2 2 WZ-type model, we find the contributions to the mass m 16ρ J θ2 ρ2 θ2 ρ2 − 8ρ2 VF ¼ e n ð þ Þ ð þ Þðn þ PÞþ : matrix as follows: ð2ρÞ n 0 1 2 2 2 Mφφ Mφρ M ðA2Þ B φθ C B 2 2 C Mρρ M φ 0 @ ρθ A Under the condition ¼ , we obtain 2 Mθθ 3 2 2 2 3 2 0 1 e m ðρ − θ Þ 2e θm 2 3∶ − − 3g 3 2 n ¼ Vρ ¼ 2 ;Vθ ¼ ; ðA3Þ þ e m 00 3ρ 3ρ B 4ρ0 2 C B C 3 2 192γ ρ3 1 3 2 B 0 e m ð 1 0þ Þ − e m C 3 2 2 2 2 3 2 2 2 ¼ B 2ρ2 2ρ2 C: e θ m ðρ − θ Þ e θm ðρ − θ Þ B 0 0 C 4∶ − @ A n ¼ Vρ ¼ 5 ;Vθ ¼ 4 : 3 2 3 2 192γ ρ3 1 4ρ 4ρ e m e m ð 2 0þ Þ 0 − 2 2 2ρ0 2ρ0 ðA4Þ ðA12Þ It is easy to verify that the case of n ¼ 3 has no solution. ρ θ In the n ¼ 4 case, the equations are satisfied when Therefore, we can stabilize and in the presence of the ρ ¼ θ ≡ ρ0. The masses are given by quartic couplings when the latter take values larger than m. We can also decouple the masses of the dilaton and the 2 3 2 V ∼ m2 V ∼ g2 ≫ 2 mm3g e m axion from F and impose the condition D mφ ¼ þ ; ðA5Þ V ∼ m2 in this case. As regards inflation, the mechanism 4ρ0 2 F discussed in the main text (Sec. V) can be applied here too. 3 2 2 e m 2 0 m− ¼ 2 ;mþ ¼ ; ðA6Þ ρ0 APPENDIX B: HYBRID SOLUTION

2 1 Since the problem of suppressing the V term comes where m are the masses of pffiffi ðρ θÞ. Hence, the vacuum F 2 from the exponential in is not stabilized in this case. In contrast to the KKLT case, supersymmetry is broken J ¼ −3 logð−CeCÞðB1Þ in the vacuum because (in terms of the canonically normalized inflaton φ,wehave pffiffi 2φ m − 3 FΦ ∼ : ðA7Þ C ¼ e ), we can change the J function to the first term ρ0 only as

075012-8 MINIMAL STAROBINSKY SUPERGRAVITY COUPLED … PHYS. REV. D 101, 075012 (2020)   1 3 2 J ¼ −3 log ð−CÞðB2Þ ≡ 3 2 ≡ −λ μ ω 2 A g y0 and B þ 2 y0 þ 4 y0 : ðB11Þ and generate the second term in (B1), leading to the As an example, the signs of the parameters can be fixed constant driving inflation in the D-type as ω < 0 and λ; μ > 0. The points y0 2 lead to a Minkowski Starobinsky potential and responsible for the instability due ð Þ vacuum with V ¼ 0, so that in this case hDi¼0 breaks to the F term, by the alternative FI term, schematically as D 2 the requirement (i). To exclude y0ð2Þ, we can impose the ∼ð1=C þ ξÞ . The parameter ξ can be fixed as ξ ¼ g in 2 12λ ω > μ y0 2 order to keep the standard Starobinsky potential. condition j j when ð Þ becomes imaginary. As regards the dilaton-axion coupled model, we have to Then the remaining minimum at y0ð1Þ is unique with the demand the following conditions: (i) the D-term potential “plus” branch according to Eq. (B8), should not cross zero between the start of inflation and the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! vacuum (otherwise, the action will become singular due to μ 36 y0 ¼ 1 þ 1 þ λjωj ; ðB12Þ the alternative FI term) and (ii) the masses of dilaton and 9jωj μ2 axion must be higher than the inflationary (Hubble) scale during inflation. where we have renamed y0ð1Þ to y0. Let us introduce the dilaton-axion pair with the following Unfortunately, the mass of ϕ vanishes in the vacuum, no-scale Kähler potential and the WZ superpotential:12 similarly to the model studied in Appendix A, so that we have to introduce the quartic couplings again. J ¼ −3 log ð−CÞ − logðΦ þ Φ¯ Þ; ðB3Þ APPENDIX C: THE Δ-DEFORMATION W λ μΦ ωΦ2 ¼ þ þ ; ðB4Þ Here we demonstrate that the introduction of a small Δ in Eq. (46) does not affect the considerations of Sec. V. where we parametrize Φ as First, let us compute the impact of Δ ≠ 0 on the scalar pffiffi potential during inflation. Equation (40) gets modified as Φ ¼ y=2 þ iθ;y¼ e− 2ϕ: ðB5Þ 1 1 3kðC−C0Þ kΔðC−C0Þ J ∼ ΔðC − C0Þ − log ðe − ξ0e Þ The scalar potential of the model is given by k 3 − Δ   1 1 3 2 ðC1Þ V ¼ −λ þ μy þ ωy2 þ ωθ2 F ð−CÞ3y 2 4 for jCj ≫ 1. Under the assumption jCΔj ≪ 1, while keep-  ing jCj ≫ 1, it reduces to þðωy − μÞ2θ2 ; ðB6Þ 1 −ξ ≡ − 0 J∞ log 3 ðC2Þ 9g2 1 2 k V ¼ ð þ 1Þ : ðB7Þ D 2 C that is exactly the same as Eq. (41), so that there is no effect of Δ. As regards another relevant function PðCÞ,a The critical points can be found analytically as straightforward calculation yields

kJ∞ −kJ∞ θ0 ¼ 0; ðB8Þ 3 − 2ξ0e e P∞ ¼ − 2Δ ðC3Þ kJ∞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kξ0e kξ0 μ 36 y0 1 − 1 1 − λω ; ≫ 1 Δ ð Þ ¼ 9ω μ2 for jCj . Here we have the small correction due to but sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! it is negligible. Thus, the key observation that both J and P μ 12 become constants for jCj ≫ 1 is unchanged, so that the y0 2 − 1 1 λω ; Δ Δ ð Þ ¼ 9ω þ μ2 ðB9Þ discussion without in Sec. V can be applied as long as is small like jCΔj ≪ 1. We conclude that the small value of rffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Δ does not affect inflation. 1 4B Next, let us consider the impact of Δ on the vacuum. Once C0 ¼ − 1 þ 1 þ ; ðB10Þ 2 A Δ ≠ 0 is included, the original vacuum (ρ ¼ ρ0, θ ¼ 0, and C ¼ −1) gets shifted. It is difficult to obtain an analytic where solution to the vacuum conditions. Therefore, we expand the scalar potential around the original vacuum as follows: 12 When at least one of the parameters of the superpotential 1 i i j vanishes, we find it impossible to obey the conditions (i) and (ii) V ¼ V0 þ V Φ þ V Φ Φ þ; ðC4Þ simultaneously. i 2 ij

075012-9 ALDABERGENOV, AOKI, and KETOV PHYS. REV. D 101, 075012 (2020) where Vi and Vij are the first and the second derivatives with Finally, the cosmological constant V0 is given by respect to all scalar fields Φi ¼ðC; ρ; θÞ, respectively, which ρ ρ θ 0 −1 are evaluated at ¼ 0, ¼ , and C ¼ , while V0 is the   2 2 3−2Bρ0 2 2 2 Jð−1Þ−2Bρ0 cosmological constant. A B e 2 g A B e V0 ¼ − þ Δ þ : The leading terms of Vρρ, Vθθ, and VCC are the same as kJð−1Þ 6ρ0 16ρ0 18ρ0ð3 þ ξ0e Þ in Eq. (44). As regards the remaining terms, we find their contributions of the order ðC8Þ

2 Vρ ¼ OðΔ Þ;VC ¼ OðΔÞ;VCρ ¼ OðΔÞðC5Þ Though the obtained correction due to Δ does uplift the and AdS vacuum, it is apparently insufficient to get a dS vacuum because the value of Δ is supposed to be small as Δ ≪ 1 Vθ ¼ Vθρ ¼ VθC ¼ 0: ðC6Þ j j , while we have the hierarchy of the parameters jΔj ≪ A

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