Massive vector multiplet inflation with Dirac-Born-Infeld type action
aa r X i v : . [ h e p - t h ] J un WU-HEP-15-9KEK-TH-1819
Massive vector multiplet inflation with Dirac-Born-Infeld type action
Hiroyuki Abe ∗ , Yutaka Sakamura , † and Yusuke Yamada ‡ Department of Physics, Waseda University,Tokyo 169-8555, Japan KEK Theory Center, Institute of Particle and Nuclear Studies, KEK,Tsukuba, Ibaraki 305-0801, Japan Department of Particles and Nuclear Physics,SOKENDAI (The Graduate University for Advanced Studies),Tsukuba, Ibaraki 305-0801, Japan
Abstract
We investigate the inflation model with a massive vector multiplet in a case that the action of the vectormultiplet is extended to the Dirac-Born-Infeld (DBI) type one. We show the massive DBI action in 4 dimensional N = 1 supergravity, and find that the higher order corrections associated with the DBI-extension make thescalar potential flat with a simple choice of the matter couplings. We also discuss the DBI-extension of the newminimal Starobinsky model, and find that it is dual to a special class of the massive DBI action. Cosmic inflation [1, 2, 3] is a plausible solution for theproblems of the standard Big Bang scenario, such as theflatness and the horizon problems. Especially, the slow-roll inflation models [4] naturally provide the primor-dial density fluctuation, which is a source of the largescale structure of the universe. The primordial curva-ture perturbation is observed by the cosmic microwavebackground experiments, and their results support theslow-roll inflation models.It is important to embed such inflation models tothe UV complete theory, such as superstring theory orits effective theory, i.e., the supergravity (SUGRA). Sofar, many inflation models have been constructed in 4dimensional (4D) N = 1 SUGRA. In many literatures,it is assumed that the inflationary potential is providedby the so-called F-term potential. However, there aresome problems in such models due to the structure ofSUGRA, such as the η problem in SUGRA, which iscaused by the factor e K in the F-term potential where K denotes the K¨ahler potential. One of the possiblesolution for the problems was suggested in Ref. [5] anddeveloped in Ref. [6]. In those models, the existence of at least two chiral multiplets are required to realizeinflation, and the multiplet other than the inflaton mul-tiplet should have a sufficiently large quartic couplingin the K¨ahler potential, otherwise it becomes tachy-onic or light, which may lead to an instability of theinflationary trajectory or an unacceptably large non-Gaussianity [7]. The solution for such a problem isthe realization of inflation without the additional chiralmultiplet, that is, the inflation with a single multiplet.Such a model was first proposed in Ref. [8], and recentlydeveloped in Ref. [9].The other interesting solution was suggested inRefs. [10, 11, 12]. In these models, the inflaton mul-tiplet is a massive U(1) vector multiplet [13, 14], whichis equivalent to a combination of Stueckelberg chiraland anti-chiral multiplets and a massless vector multi-plet. The inflation is driven by the scalar componentof the massive vector multiplet. One of the advantagesof such a model is that only the single massive vectormultiplet is the sufficient source for inflation, and the η problem is absent because the inflaton potential isprovided by the so-called D-term potential. Therefore,the inflation is realized in a relatively simple way.From the theoretical viewpoint, it is important to ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] Especially, it was found that the DBI action in N = 1 4D global SUSY naturally appears from the par-tial N = 2 SUSY breaking [19, 20], which is realized byimposing a condition between two N = 1 multiplets in-cluded in an N = 2 vector multiplet. The condition canalso be embedded into SUGRA [23, 24], which realizesthe DBI action in 4D N = 1 SUGRA. The matter cou-pled DBI action was also discussed in 4D N = 1 globalSUSY [25] and SUGRA [23, 24]. Recently, in SUGRA,we developed the matter coupled DBI type action in-cluding matters charged under the corresponding U(1)symmetry in Ref. [26].To explore the UV completions of the massive vec-tor multiplet inflation models, we will extend the actionof the massive vector multiplet to the DBI type action.Such an action can be constructed by the DBI actionof a massless vector multiplet coupled to the Stueckel-berg multiplet, which non-linearly transforms under thecorresponding U(1) symmetry. The DBI action withcharged matter multiplets was studied in our previouswork [26] as mentioned above. Such a DBI-extendedaction contains higher order terms of the matter fields,associated with supersymmetric higher derivative cou-plings. We call them the DBI corrections throughoutthis paper. We found that the DBI corrections signifi-cantly modify the D-term potential. As we will show,the DBI corrections become important if the cut-offscale associated with the DBI action is smaller thanthe Planck scale. Contrary to a naive expectation, thecorrections make the scalar potential flatter. This is aninteresting feature of the DBI-extension. We demon-strate such a feature for two concrete models, whichare the chaotic [4] and the Starobinsky model [2].In Ref. [27], it is shown that the Starobinsky modelin the new minimal SUGRA is dual to the massive vec-tor multiplet action. We will also investigate the DBI-extension of the former, and find that it is dual to a special class of the massive DBI action, which has a re-stricted form of the matter couplings. Because of therestricted form, the flatness of the inflaton potential isno longer protected under the DBI corrections.The remaining parts of this paper are organized asfollows. In Sec. 2, we briefly review the massive vectoraction in SUGRA and its application to the inflationmodel based on Ref. [11]. Sec. 3 is devoted to the con-struction of the DBI action of a massive vector multi-plet. Then we obtain the modified D-term potential.In Sec. 4, we discuss how the DBI corrections affect theinflaton potential, and find that the potential becomesflatter when the cut-off scale of the DBI sector is muchsmaller than the Planck scale. We also discuss the DBI-extension of the Starobinsky model in Sec. 5, and showthe dual action of it. Finally, we conclude in Sec. 6. InAppendix A, we give a brief review of the Starobinskymodel in the new minimal SUGRA. In this section, we briefly review the massive vec-tor action in supergravity and the inflation modelswith it, based on Ref. [11]. The action in conformalSUGRA [28, 29] is given by S = (cid:20) S ¯ S Φ(Λ + ¯Λ + gV ) (cid:21) D −
14 [ W α ( V ) W α ( V )] F (1)where S is the chiral compensator, Λ is a Stueckelbergchiral multiplet, V is a vector multiplet, g is the gaugecoupling, Φ( x ) is an arbitrary real function of x , W α isthe field strength supermultiplet, and [ · · · ] D,F are thesuperconformal D- and F-term density formulae respec-tively [29]. The supergauge transformations of Λ and V are Λ → Λ + g Σ , V → V − Σ − ¯Σ , (2)where Σ denotes the gauge parameter chiral multiplet.By choosing Σ = − Λ /g , the vector multiplet V becomesa massive vector multiplet , and the action (1) becomes S = (cid:20) S ¯ S Φ( gV ) (cid:21) D −
14 [ W α ( V ) W α ( V )] F . (3) See also Refs.[17, 22] for reviews. The action (1) is also dual to the massive tensor action as shown in Refs. [11, 30], however we do not discuss it further. We can also regard the action (1) as the ordinary supergravity action with the K¨ahler potential K = − − Φ(Λ + ¯Λ + gV ) / For notational simplicity, we choose Φ(Λ + ¯Λ + gV ) = − e − J/ , where J = J ( (Λ+ ¯Λ+ gV )). After imposingthe conventional superconformal gauge conditions [31]and integrating out auxiliary fields, we obtain the fol-lowing bosonic Lagrangian L B in the Einstein frame, L B = − J ′′ ( C ) ∂ µ C∂ µ C − g J ′′ (cid:18) A µ − g ∂ µ θ (cid:19) − g J ′ ( C )) − F µν F µν + 12 R, (4)where C = ReΛ, θ = ImΛ, the prime on J denotes thederivatives with respect to C , A µ is the vector com-ponent of V , F µν is the field strength of A µ , and R denotes the Ricci scalar. By choosing the U(1) gaugeas θ = 0, the second term in Eq. (4) becomes the massterm of A µ with m A = g J ′′ ( C ).In the following, we identify C as the inflaton withits scalar potential given by V = g J ′ ( C )) . (5)We can reconstruct various types of the inflaton po-tential by choosing the function J ( C ). For example,the choice J = C / V = g C / J ( C )takes a special form given by J = − log( − Ce C / C is L = − C ∂ µ C∂ µ C − g C ) . (6)In terms of the canonical scalar φ = p / − C ), theLagrangian (6) can be rewritten as L = − ∂ µ φ∂ µ φ − g − e − √ / φ ) . (7) We extend the massive vector action reviewed inSec. 2 to the DBI type one developed in our previous work [26]. Let us consider the following action, S = (cid:20) S ¯ S Φ(Λ + ¯Λ + gV ) (cid:21) D − [ hS X ( W , ¯ W )] F . (8)In the second term, X ( W , ¯ W ) is a solution of the fol-lowing equation, S X = W α W α − X Σ c ( ω (Λ , ¯Λ) ¯ S ¯ X ) , (9)where X is a chiral multiplet, Σ c ( · ) denotes the chiralprojection operator in conformal SUGRA acting on themultiplet in its argument, h is a real constant parame-ter, and ω (Λ , ¯Λ) is a real function of Λ and ¯Λ.After the superconformal gauge fixings and inte-grating out the auxiliary fields, we obtain the followingbosonic action, S = Z d x √− g " R − J ′′ ( C ) ∂ µ C∂ µ C − g J ′′ A µ A µ + Z d x e J hω − p ˜ P r − det (cid:16) g µν + 2 e − J √ ω F µν (cid:17)! , (10)where we have chosen the U(1) gauge condition as θ = 0, and ˜ P ≡ ωe − J/ g ( J ′ ( C )) h . (11)In the following discussion, let us consider a simple casewith h = 1 / ω = exp(4 J/ / (4 M ), where M is apositive constant. Then the action (10) becomes S = Z d x √− g " R − J ′′ ( C ) ∂ µ C∂ µ C − g J ′′ A µ A µ + Z d xM − √ P s − det (cid:18) g µν + 1 M F µν (cid:19)! , (12)where P ≡ g ( J ′ ( C )) M . (13) We use the Planck unit convention in which M pl ∼ . × GeV = 1. For a detailed derivation of the action, see Ref. [26].
3n the limit of M → ∞ , the action (12) reduces to theaction (4). Indeed, in this limit, we easily find S = Z d x (cid:20) − g ( J ′ ( C )) − F µν F µν + O ( M − ) + · · · (cid:21) (14)where the ellipsis denotes terms independent of M . Wecan also see this by noticing that, in such a limit, Eq. (9)becomes S X = W α W α . Then, the action (8) becomesexactly the same as one in Eq. (1).Therefore, we can regard the action (12) as the DBItype extension of a massive vector multiplet, and M asa cut-off scale of the DBI sector. Note that the massiveDBI action loses the gauge invariance which the origi-nal one [15] has, due to the spontaneous breaking of theU(1) symmetry. Such a spontaneous breaking occurs,e.g. in superstring models when the Green-Schwartzanomaly cancellation [33] is realized, which can be de-scribed by the Stueckelberg multiplet as in Ref. [34]. In the previous section, we derived the DBI action of amassive vector multiplet. In this section, let us discussthe effects of the DBI corrections on the scalar poten-tial, and its implication for inflation models.From Eq. (12), we obtain the DBI-extended scalarpotential of C , V mod = M ( √ P − M r g ( J ′ ( C )) M − ! . (15)In Ref. [35], it is concluded that in inflation modelswith a massive vector multiplet, the higher order cor-rections associated with SUSY higher derivative termsare negligibly small in general. Indeed, all the DBI cor-rections in Eq. (15) are proportional to the couplingconstant g n /M n − ( n ≥ g ( J ′ ( C )) / g should be much smaller than 1 to explain the ampli-tude of the primordial density perturbation.However, we should notice that, even in the case of g ≪
1, the higher order corrections become importantif M is also much smaller than 1. In fact, a param-eter that controls the DBI corrections is β ≡ g /M ,rather than g . In terms of β , we rewrite the scalar potential (15) as V mod = g β ( p β ( J ′ ( C )) − . (16)We find that for sufficiently large β the scalar potentialis approximately given by V ∼ ( g / √ β ) | J ′ ( C ) | .In general, when the cut-off scale M is low, thehigher order corrections ruin the flat profile of the scalarpotential. Contrary to such a naive expectation, wenotice that in the regime of β ≫
1, the DBI correc-tions make the shape of the scalar potential flatter thanthat of the ordinary D-term potential (5), given by V ∼ ( J ′ ( C )) .As a concrete example, let us consider the case with J = C /
2, which corresponds to the simplest chaoticinflation model mentioned in Sec. 2. Then the scalarpotential is given by V = g β ( p βC − . (17)We show the form of the scalar potential with variousvalues of β in Fig. 1. - - VV Β= Β= Β= - Β= - Figure 1: The forms of the scalar potential (17) withdifferent values of β are shown. The potential is nor-malized at C = 10 ( V ≡ V ( C = 10)).In Fig. 1, the scalar potential asymptotically convergeson the linear type potential for larger β , which is ex-pected from Eq. (17).Due to the modification of the scalar potential, thepredicted values of cosmological parameters are also dif-ferent from ones of the original quadratic potential if β is sufficiently large. We numerically calculated thespectral index n s and the tensor to scalar ratio r at the4orizon exit. We obtained following sets of values for β = 10 − and β = 10 respectively,( n s , r ) = ( (0 . , . β = 10 − )(0 . , . β = 10) , (18)where we show the values at N = 60, and N is the num-ber of e-foldings. In both cases, g should be O (10 − ).For β = 10 − , the cut-off scale M is O (1), and thevalues of the cosmological parameters are not affectedby the DBI corrections, which is compatible with theconclusion of the Ref. [35]. On the other hand, the pre-dicted parameters are altered by the DBI correctionsfor β = 10. In this case, M ∼ − , which is higherthan the inflation scale H ∼ − .As another example, let us discuss the Starobinskymodel, shown in Sec. 2. In that case, J ( C ) is given by J = − log( − Ce C / C isgiven by L = − C ∂ µ C∂ µ C − g β r β C ) − ! . (19)We redefine C as C = − exp( p / φ ), and then we canrewrite the action (19) as L = − ∂ µ φ∂ µ φ − g β r β − e − √ / φ ) − ! . (20)We show the scalar potential in Eq. (20) with variousvalues of β in Fig. 2. Φ VV Β= Β= Β= - Β= - Figure 2: The forms of the scalar potential in Eq. (20)with different values of β are shown. The potential isnormalized at φ = 7 ( V ≡ V ( φ = 7)). As in the previous model, we obtained the followingvalues of ( n s , r ) for β = 10 − and β = 10,( n s , r ) = ( (0 . , . β = 10 − )(0 . , . β = 10) , (21)where we have shown the values at N = 60. Asshown in Fig. 2, the scalar potential in Eq. (20) isnot much affected by the DBI corrections. The rea-son can be understood as follows. In the region thatexp( − p / φ ) ≪
1, the original potential in Eq. (7)can be approximately written as V = 9 g − e − √ φ ) ∼ g − e − √ φ ) . (22)On the other hand, for a sufficiently large β , the scalarpotential in Eq. (20) can be approximated as V ∼ g β r β − e − √ φ ) = 3 g √ β (1 − e − √ φ ) . (23)Therefore, the form of the leading order terms is notaffected by the DBI corrections, and therefore the pre-dicted cosmological parameters (21) are almost un-changed. R model in newminimal SUGRA As we mentioned in Sec. 1, the Starobinsky model inthe new minimal SUGRA is dual to the massive vec-tor inflation model with J = − log( − Ce C /
3) that wetreated in Sec. 2. We review the Starobinsky model inthe new minimal SUGRA in Appendix A.Since the dual of the new minimal Starobinskymodel is described by a massive vector multiplet, it isinteresting to construct the DBI-extension of the newminimal Starobinsky model. As we find below, it isdual to the massive DBI action with a special form of ω . As shown in Eq. (38), the Starobinsky model in thenew minimal SUGRA consists of a real linear compen-sator L and a real multiplet V R = log( L / |S| ) where S is a chiral multiplet. As is the case with gauge mul-tiplets, we can construct the following DBI type actionin the new minimal Starobinsky model, S = (cid:20) L V R (cid:21) D + [ − γX ( W , ¯ W , L )] F , (24)5here γ is a real constant, and X ( W , ¯ W , L ) is a so-lution of the following equation (see Appendix A inRef. [26]), X = W α ( V R ) W α ( V R ) − κ Σ c ( | X | L − ) , (25)where κ is a positive constant. Since it is difficult tosolve Eq. (25), we rewrite the action in Eq. (24) with achiral Lagrange multiplier multiplet M as S = (cid:20) L V R (cid:21) D + [ − γX ] F + [2 M ( W ( V R ) − κ Σ c ( | X | L − ) − X )] F . (26)After the gauge fixings of V R and the superconformalsymmetry, the bosonic Lagrangian of the action (26)becomes L| B = 12 R + 34 B µ B µ − A ( R ) µ B µ + 4 κλ | F X | − λD R ) + 2 λ F ( R ) µν F ( R ) µν − iχ F ( R ) µν ˜ F ( R ) µν − ( γ + 2 M ) F X − ( γ + 2 ¯ M ) ¯ F ¯ X , (27)where F X is the F-term of X , λ ≡ Re M , χ ≡ Im M , D ( R ) ≡ ( R + 3 B µ B µ / / A ( R ) µ is the vector compo-nent of V R , B µ is an auxiliary vector component of L ,and F ( R ) µν = ∂ µ A ( R ) ν − ∂ ν A ( R ) µ . Integrating out F X , λ , χ , we obtain the following action. L| B = 12 R + 34 B µ B µ − A ( R ) µ B µ − γκ + γκ q κD R ) − κ F ( R ) µν F ( R ) µν + κ ( F ( R ) µν ˜ F ( R ) µν ) = 12 R + 34 B µ B µ − A ( R ) µ B µ − γκ + γκ q κD R ) − det( η ab + √ κ F ab ) . (28)The action (28) has the DBI type structure with re-spect to the vector A ( R ) µ , but it also includes D R ) =( R + 3 B µ B µ / in the square root, which yields thehigher curvature correction [36, 37].Next, let us discuss the dual action of (28). As per-formed in Appendix A, we can derive the dual action byusing a real linear multiplier multiplet U and a gaugemultiplet V . Let us consider the following action. S = (cid:20) L V R (cid:21) D + [ − γX ] F + [2 M ( W ( V ) − κ Σ c ( | X | L − ) − X )] F + [ U ( V − V R )] D . (29) The equation of motion of U gives a solution with re-spect to V shown in Eq. (41), which reproduces theaction (26). On the other hand, if we solve it with re-spect to L , which is shown in Eq. (42), the action (29)becomes S = (cid:20) |S| (Λ + ¯Λ + V ) exp(Λ + ¯Λ + V ) (cid:21) D + [ − γX ] F + [2 M ( W ( V ) − κ Σ c ( | X | |S| − e − V ) ) − X )] F . (30)By the field redefinitions Λ → Λ / , V → gV / , S → S / √ , M → M /g , X → g X/
4, we obtain S = h a | S | Ce C i D + (cid:20) − g γ X (cid:21) F + [2 M ( W ( V ) − κg c ( | X | | S | − e − C ) − X )] F , (31)where C = (Λ+ ¯Λ+ gV ) /
2. We choose parameters γ and κ as γ = g − , κ = 3 g − M − , and then the action (31)becomes S = Z d x √− g " R − J ′′ ( C ) ∂ µ C∂ µ C − g J ′′ A µ A µ + Z d x M C − √ P s − det (cid:18) g µν + CM F µν (cid:19)! , (32)where P = 1 + g C ( J ′ ( C )) M . (33)The action (32) is a special case of the DBI-extended Starobinsky model, i.e., Eq. (10) with J = − log( − Ce C / ω as ω = 9 e − C M . (34)Note that this choice is different from the one in theprevious sections: ω = e J M = 9 e − C M C , (35)which is chosen so that the coefficient of F µν in thesquare root in (10) becomes a constant, just like the or-dinary DBI action. Namely, although Eq.(10) with J =6 log( − Ce C /
3) and with (35) is the DBI-extension ofthe dual action to the Starobinsky model, it is not thedual action to the DBI-extension of the (new minimal)Starobinsky model, which is given by (32).In terms of the canonical scalar field φ = p / − C ), we can obtain the following scalar po-tential. V = g β e − √ φ s βe √ φ − e − √ φ ) − , (36)where β = g /M . For comparison, we show the scalarpotential (36) with different values of β and the originalone in the Starobinsky model (7) in Fig. 3. Φ VV Β= - Β= - Β= - R model Figure 3: The forms of the scalar potential (36) withdifferent values of β and the original one in Eq. (7)are shown. The potential is normalized at φ = 7( V ≡ V ( φ = 7)).From Fig. 3, we find that the scalar potential in Eq. (36)changes drastically even with a small β , unlikely to theone in Eq. (20) discussed in Sec.4. We can understandthis property as follows. For A ( R ) µ = 0, the auxiliaryfield B µ is integrated out and we obtain B µ = 0, thenthe Lagrangian (28) becomes L = 12 R + g β ( p βR − . (37)If βR ≫
1, the structure ∼ R + R does not appear,that means the Starobinsky model appears as the lowcurvature (compared with 1 / √ β ) approximation in theDBI-extension of a massive vector action dual to theone in new minimal SUGRA. In this case, the confor-mal symmetry, which the term R has, is severely bro-ken. Thus, the inflaton potential loses its flat profile. We remark that, from the viewpoint of superconformalformulation, the essential difference between the action(12) in section 3 and the one (32) in this section is thechoice of the function ω in Eq. (9). We have constructed the DBI action of a massive vectormultiplet in 4D N = 1 SUGRA, and also shown howthe DBI corrections affect the inflationary potential inthe massive vector inflation model. Such an extensionmay be required to realize the model in superstring the-ory if the action is an effective action of D-brane. Asshown in Sec. 3, the DBI action of a massive vectormultiplet is obtained from the massless one coupled tothe Stueckelberg chiral multiplet, which is based on theresult in our previous work [26]. As a nontrivial con-sequence of the DBI-extension, the D-term potential ismodified.It is a common expectation that the higher ordercorrections ruin the flatness of the scalar potential.However, we have found that the modified scalar poten-tial is flatter than the one before the DBI-extension, asshown in Sec. 4. We have also shown the concrete exam-ples, the quadratic chaotic inflation and the Starobinkyinflation. Especially for the former case, the scalar po-tential is drastically modified into the linear one, evenwith the cut-off scale M larger than the Hubble scaleduring inflation. Such an interesting feature of theDBI action may be important for inflation models inSUGRA. The feature is also favored by the latest re-sults from the CMB observation by the Planck satel-lite [38], because the DBI correction reduces the valueof r (see examples in Sec.4). It is also worth to notethat the functions J ( C ) and ω (Λ , ¯Λ) are not so arbi-trary, restricted by geometries and symmetries of thebackground, if it is realized in superstring theory. Ifthe simplest case given in Eq. (12) is realized, the DBIcorrection discussed in this work always flatten the po-tential as shown below Eq. (16).We have also discussed the DBI-extension of theStarobinsky model in the new minimal SUGRA. Theaction (28) is a possible extension of the new minimalSUGRA, which has higher order terms of R withoutghosts. As shown in Sec. 5, the action is dual to aspecial class of the massive DBI action, which is differ-ent from the simple massive DBI action (12). In thatcase, the scalar potential is highly affected by the DBIcorrections and it loses the flat profile because of the7reaking of the conformal symmetry, which the origi-nal Strarobinsky model has during inflation. Acknowledgments
H.A., Y.S. and Y.Y. are supported in part by Grant-in-Aid for Young Scientists (B) (No. 25800158), Grant-in-Aid for Scientific Research (C) (No.25400283), and Re-search Fellowships for Young Scientists (No.26-4236),which are from Japan Society for the Promotion of Sci-ence, respectively.
A The Starobinsky model in newminimal SUGRA
We briefly review the Starobinsky model in the newminimal SUGRA [32]. From the conformal SUGRAviewpoint, the new minimal SUGRA can be regardedas conformal SUGRA with a real linear compensatormultiplet denoted by L .Supermultiplets in conformal SUGRA are charac-terized by the Weyl and the chiral weights denoted by w and n respectively, which are the charges for the di-latation and U(1) A . L has the weights ( w, n ) = (2 , S = (cid:20) L V R (cid:21) D + [ − γ W α ( V R ) W α ( V R )] F , (38)where V R = log( L / |S| ), and S is a chiral multipletwith ( w, n ) = (1 , γ is a positive constant. Itis worth noting that a linear multiplet has a prop-erty [ L (Λ + ¯Λ)] D = 0, where Λ is a chiral multipletwith ( w, n ) = (0 , S → S e Σ . Under thistransformation, V R transforms as V R → V R − Σ − ¯Σ likea gauge multiplet shown in Eq. (2). Therefore, we canperform the “gauge” transformation to remove S fromthe physical action, and we choose the gauge in which S = 1. After the superconformal gauge fixings, thebosonic part of the Lagrangian L in Eq. (38) can becalculated as L| B = 12 R + 2 γ R − γ F ( R ) µν F ( R ) µν − A ( R ) µ B µ + (cid:18)
34 + 2 γ R (cid:19) B µ B µ + γ ( B µ B µ ) , (39) where A ( R ) µ is the vector component of V R , F ( R ) µν = ∂ µ A ( R ) ν − ∂ ν A ( R ) µ , and B µ is an auxiliary vector com-ponent of L . On the trajectory A ( R ) µ = 0, we canintegrate B µ out, and obtain the usual Starobinsky La-grangian L = R/ γR / V R transforms like a U(1) gauge multiplet, itis natural to think that the action (38) has a dual formwith a gauge multiplet. Indeed, the action (38) can berewritten as S = (cid:20) L V R (cid:21) D + [ − γ W α ( V ) W α ( V )] F + [ U ( V − V R )] D (40)where U is a real linear multiplet, and V is a gaugemultiplet. By varying U , we obtain V = V R − Λ − ¯Λwhere Λ is a chiral multiplet. Substituting this intoEq. (40), we can reproduce the action (38). On theother hand, we can also solve the equation of motion of U with respect to L as follows. V R = log (cid:18) L |S| (cid:19) = (Λ + ¯Λ + V ) , (41)equivalently, L = |S| exp (cid:0) Λ + ¯Λ + V (cid:1) . (42)Thus, we obtain the following dual action, S dual = (cid:20) |S| (Λ + ¯Λ + V ) exp(Λ + ¯Λ + V ) (cid:21) D + [ − γ W α ( V ) W α ( V )] F . (43)After field redefinitions Λ → Λ / , V → gV / , S → S / √
3, we obtain S dual = (cid:20) | S | (cid:18)
12 (Λ + ¯Λ + gV ) (cid:19) exp (cid:18)
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