Higgs Mechanism and Cosmological Constant in N = 1 Supergravity with Inﬂaton in a Vector Multiplet

Eur. Phys. J. C (2017) 77:233 DOI 10.1140/epjc/s10052-017-4807-8

Letter

Higgs mechanism and cosmological constant in N = 1 supergravity with inﬂaton in a vector multiplet

Yermek Aldabergenov1,a, Sergei V. Ketov1,2,3,b 1 Department of Physics, Tokyo Metropolitan University, Minami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan 2 Kavli Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Chiba 277-8568, Japan 3 Institute of Physics and Technology, Tomsk Polytechnic University, 30 Lenin Ave., Tomsk 634050, Russian Federation

Received: 2 March 2017 / Accepted: 3 April 2017 © The Author(s) 2017. This article is an open access publication

Abstract The N = 1 supergravity models of cosmologi- sive vector multiplet were constructed in Ref. [6] by exploit- cal inflation with an inflaton belonging to a massive vector ing the non-minimal self-coupling of a vector multiplet to multiplet and spontaneous SUSY breaking after inflation are supergravity [7]. The supergravity inflationary models [6] reformulated as the supersymmetric U(1) gauge theories of have the single-field scalar potential given by an arbitrary real a massless vector superfield interacting with the Higgs and function squared. Those scalar potentials are always bounded Polonyi chiral superfields, all coupled to supergravity. The from below and allow any desired values of the CMB observ- U(1) gauge sector is identified with the U(1) gauge fields of ables ns and r. However, the minima of the scalar potentials the super-GUT coupled to supergravity, whose gauge group of [6] have a vanishing cosmological constant and a vanish- has a U(1) factor. A positive cosmological constant (dark ing Vacuum Expectation Value (VEV) of the auxiliary field energy) is included. The scalar potential is calculated, and its D, so that they allow only Minkowski vacua where super- de Sitter vacuum solution is found to be stable. symmetry is restored after inflation. A simple extension of the inflationary models [6]was proposed in Ref. [8] by adding a Polonyi (chiral) multiplet 1 Introduction [9] with a linear superpotential. The inflationary models [8] also accommodate arbitrary values of ns and r, and have a PLANCK observations [1Ð3] of the Cosmic Microwave Minkowski vacuum after inflation, but with spontaneously Background (CMB) radiation favor chaotic slow-roll infla- broken supersymmetry (SUSY). In this paper we further tion in its single-field realization, i.e. the large-field inflation extend the models of Ref. [8] by allowing them to have a driven by a single scalar called inflaton with an approximately positive cosmological constant, i.e. a de Sitter vacuum after flat scalar potential. inflation. Embedding inflationary models into N = 1 four- Yet another motivation comes from an exposition of the dimensional supergravity is needed to connect them to par- super-Higgs effect in supergravity by presenting the new ticle physics theory beyond the Standard Model of elemen- U(1) gauge-invariant form of the class of inflationary mod- tary particles and to quantum gravity. Most of the literature els under investigation. This paves the way toward embed- about inflation in supergravity is based on an assumption ding our models into the superymmetric Grand Unification that the inflaton belongs to a chiral (scalar) multiplet Ð see Theories (sGUT) coupled to supergravity, when they have a e.g., the reviews [4,5]. However, the inflaton can also be spontaneously broken U(1) factor in the sGUT gauge group. assigned to a massive N = 1 vector multiplet. It has some The physical scale of cosmological inflation can be iden- theoretical advantages because there is only one real scalar tified with the Hubble (curvature) scale H ≈ 1014 GeV 13 in an N = 1 massive vector multiplet. The η-problem does or the inflaton mass minf ≈ 10 GeV. The inflationary not arise because the scalar potential of a vector multiplet in scale is thus less (though not much less!) than the sGUT supergravity is of the D-type instead of the F-type. The mini- scale of 1016 GeV. The simple sGUT groups SU(5), SO(10) mal inflationary models with the inflaton belonging to a mas- and E6 are well motivated in the CalabiÐYau compactified heterotic strings, however, they usually come with at least ( ) a e-mail: [email protected] one extra “undesired” U 1 factor in the gauge group. The b e-mail: [email protected] well-known examples include the gauge symmetry break- 123 233 Page 2 of 6 Eur. Phys. J. C (2017) 77:233 ing E6 → SO(10) × U(1), SO(10) → SU(5) × U(1), 1 2 K +2J −1 V = J + e K ∗ (Wi + Ki W)(W j + K j∗ W) and the “flipped” SU(5) × U (1) sGUT originating from 2 ij X heterotic strings. Exploiting the Higgs mechanism in super- J 2 ( ) − 3 − 2 WW , (3) gravity allows us to propose an identification of the U 1 J gauge vector multiplet of those sGUT models with the infla- ≡ ton vector multiplet we consider, thus unifying inflation with where we have introduced the vierbein determinant e a those sGUT in supergravity. Besides the sGUT gauge uni- detem, the spacetime scalar curvature R, the complex scalars fication, related proton decay and baryon number violation, Ai as physical components of i ; the real scalar C and having the U(1) factor in the sGUT gauge group allows one the real vector Bm, with the corresponding field strength = D − D to get rid of monopoles, because the gauge group is not semi- Fmn m Bn n Bm, as physical components of V .The W simple [10]. Having a positive cosmological constant takes functions K , J and now represent the lowest components (Ai and C) of the corresponding superfields. As regards into account dark energy too. ∂ K ∂ K their derivatives, we use the notation Ki ≡ ∂ , Ki∗ ≡ , Our paper is organized as follows. In Sect. 2 we briefly Ai ∂ Ai ∂2 K ∂ J ∂W ∂W review the supergravity models [8]. In Sect. 3 we present their Kij∗ ≡ , J ≡ ∂ , Wi ≡ ∂ , Wi ≡ . As is clear ∂ Ai ∂ A j C Ai ∂ Ai U(1) gauge-invariant formulation and the Higgs mechanism. from Eq. (2), the absence of ghosts requires J (C)>0, A positive cosmological constant is added in Sect. 4.The where the primes denote differentiations with respect to the scalar potential and it stability are studied in Sect. 5.Our given argument.2 conclusion is given by Sect. 6. For our purposes here, we restrict ourselves to a single chiral superfield whose Kähler potential and the superpotential are those of the Polonyi model [9]: 2 Scalar potential and SUSY breaking with a massive vector multiplet in the absence of a cosmological K = , W = μ( + β), (4) constant with the parameters μ and β. The choice (4) is quite natural The inflationary models of Ref. [8] are defined in curved (and unique) for a nilpotent (VolkovÐAkulov) chiral super- superspace of N = 1 supergravity [11] by the Lagrangian field obeying the constraint 2 = 0, though we do not 1 (MPl = 1) employ the nilpotency condition here, in order to avoid its possible clash with unitarity at high energies. 2 3 − 1 (K +2J) 1 α L = d θ2E (DD − 8R)e 3 + W Wα + W 8 4 A substitution of Eq. (4) into the Lagrangian (2) yields + . ., h c (1) −1 1 m ¯ 1 mn 1 m e L =− R − ∂m A∂ A− Fmn F − J ∂mC∂ C 2 4 2 in terms of chiral superfields i , representing ordinary (other 1 m 1 2 2 AA¯+2J ¯ 2 than inflaton) matter, with a Kähler potential K = K ( , ) − J Bm B − J − μ e |1 + Aβ+AA| i i 2 2 and a chiral superpotential W = W( ), and interact- i 2 ing with the vector (inflaton) superfield V described by a − − J | + β|2 , 3 2 A (5) real function J = J(V ) and having the superfield strength J ≡−1 (DD − R)D Wα 4 8 α V . We have also introduced the where the complex scalar A is the lowest component of the E chiral density superfield 2 and the chiral scalar curvature Polonyi chiral superfield . R superfield [11]. The Minkowski vacuum conditions, After eliminating the auxiliary fields and changing the 1 ¯+ initial (Jordan) frame to an Einstein frame, the bosonic part V = J 2 + μ2eAA 2J |1 + Aβ + AA¯|2 2 of the Lagrangian (1) reads [8] 2 J 2 −1 1 m ¯ 1 mn − 3 − 2 |A + β| = 0, (6) e L =− R − Kij∗∂m Ai ∂ A j − Fmn F 2 4 J ¯ 1 m 1 m ∂ = μ2 AA+2J ( + ¯β + ¯) + ( + β) − J ∂mC∂ C − J Bm B − V, (2) ¯ V e A 1 A AA A 2 2 A and it has the scalar potential J 2 × (1 + Aβ + AA¯) − 3 − 2 (A + β) J 1 Our notation and conventions coincide with the standard ones in J 2 Ref. [11], including the spacetime signature (−, +, +, +).TheN = 1 + | + β+ ¯|2− − | +β|2 = , A 1 A AA 3 2 A A 0 (7) superconformal calculus [6,7] after the superconformal gauge fixing J is equivalent to the curved superspace description of N = 1 Poincaré supergravity. 2 Our J-function differs by the sign from that in Refs. [6,7]. 123 Eur. Phys. J. C (2017) 77:233 Page 3 of 6 233 2 AA¯+2J ¯ 2 The U(1) gauge symmetry of the Lagrangian (9) allows us ∂C V = J J + 2μ e |1 + Aβ + AA| to choose a different (Wess–Zumino) supersymmetric gauge 2 by “gauging away” the chiral and anti-chiral parts of the gen- − − J + J J | + β|2 = , 1 2 A 0 (8) eral superfield V via the appropriate choice of the superfield J J 2 parameters Z and Z, can be satisfied when J = 0, which separates the Polonyi multiplet from the vector√ multiplet. The Polonyi√ field VEV V |=DαDβ V |=Dα˙ Dβ˙ V |=0, =( − ) β = − is then given by A 3 1 and 2 3[9]. This m Dα˙ Dα V |=σαα˙ B , solution describes a stable Minkowski vacuum with sponta- m =μ β 1 m αβ˙ n β neous SUSY breaking at an arbitrary scale F .The Dα W |= σαα˙ σ (2iFmn) + δα D, = 4 related gravitino mass√ (at the minimum having J 0) is 2− 3 16 m given by m / = μe . There is also a massive scalar of DDDDV |= b Bm + 8D, 3 2 3 mass 2m3/2 and a massless fermion in the physical spectrum. As a result, the Polonyi field does not affect the infla- where the vertical bars denote the leading field components tion driven by the inflaton scalar C belonging to the mas- of the superfields. sive vector multiplet and having the D-type scalar potential It is straightforward (but tedious) to calculate the bosonic ( ) = 1 2 V C 2 J with a real J-function. Of course, the true part of the Lagrangian in terms of the superfield components inflaton field should be canonically normalized via the proper in an Einstein frame, after elimination of the auxiliary fields field redefinition of C. and Weyl rescaling. We find

−1 1 m ¯ 1 mn e L =− R − Kij∗ ∂ Ai ∂m A j − Fmn F 3 Massless vector multiplet and super-Higgs mechanism 2 4 − ∂ ∂m ¯ − 1 m 2Jhh¯ m h h JV 2 Bm B The matter-coupled supergravity model (1) can also be con- 2 + ( ∂m − ∂m ¯) − V, sidered as a supersymmetric (Abelian, non-minimal) gauge iBm JVh h JV h¯ h (12) theory (coupled to supergravity and a Higgs superfield) in ¯ the (supersymmetric) gauge where the Higgs superfield is where h, h are the Higgs field and its conjugate. We use the ∂2 J ∂2 J ∂2 J ( ) notation J ¯ ≡ |, J ≡ | and J 2 ≡ |.As gauged away (say, equal to 1). When the gauge U 1 symme- hh ∂h∂h¯ Vh ∂h∂V V ∂V 2 try is restored by introducing back the Higgs (chiral) super- regards the scalar potential, we get field, the vector superfield V becomes the gauge superfield of ( ) V = 1 2 + K +2J ( + )IJ∗ ( + ( + ) ) a spontaneously broken U 1 gauge group. In this Section we JV e K 2J WI K 2J I W restore the gauge symmetry in the way consistent with local 2 × ( ∗ + ( + ) ∗ ) − , supersymmetry, and then compare our results with those of W J K 2J J W 3W W (13) the previous Section. where the capital Latin indices I, J collectively denote all We start with a Lagrangian having the same form as (1), chiral superfields (as well as their lowest field components) 2 3 − 1 (K +2J) 1 α including the Higgs superfield. L = d θ2E (DD − 8R)e 3 + W Wα +W(i ) 8 4 The standard U(1) Higgs mechanism setting appears after + . ., = 1 2V ¯ h c (9) employing the canonical function J 2 he h. As regards the Higgs sector, it leads to where K = K (i , j ) and the indices i, j, k refer to the −1 m ¯ ¯ m m ¯ chiral (matter) superfields, excluding the Higgs chiral super- e LHiggs =−∂m h∂ h + iBm(h∂ h − h∂ h) field, which we denote H, H. Now, in contrast to the previous −hhB¯ Bm − V. (14) section, the real function J also depends on the Higgs super- m = ( 2V ) field as J J He H , while the vector superfield V is When parameterizing h and h¯ as massless. The Lagrangian (9) is invariant under the supersymmetric U(1) gauge transformations = 1 (ρ + ν) iζ , ¯ = 1 (ρ + ν) −iζ , − h √ e h √ e (15) H → H = e iZH, H → H = ei Z H, (10) 2 2 i V → V = V + (Z − Z), (11) 2 where ρ is the (real) Higgs boson, ν ≡h=h¯ is the Higgs whose gauge parameter Z itself is a chiral superfield. The VEV, and ζ is the Goldstone boson, in the unitary gauge of → = −iζ → = +∂ ζ Lagrangian (1) of Sect. 2 is recovered from Eq. (9)inthe h h e h and Bm Bm Bm m , we reproduce 2V gauge H = 1, after the redefinition Jnew(e ) = Jold(V ). the standard result [12], 123 233 Page 4 of 6 Eur. Phys. J. C (2017) 77:233

−1L =−1∂ ρ∂mρ − 1(ρ + ν)2 m − V. Im A e Higgs m Bm B (16) 1.0 2 2 0.0 0.5 0.5 1.0 The same result is also achieved by considering the super- Higgs mechanism where, in order to get rid of the Goldstone 2 mode, we employ the super-gauge transformations (10) and ( − (11), and define the relevant field components of Z and i Z 1 Z) as 0 i m 1.0 |=ζ + ξ, Dα˙ Dα( − )|=σ ∂ ζ. 0.5 Z i Z Z αα˙ m (17) 0.0 2 0.5 Re A 1.0 Examining the lowest components of the transformation (10), ˜ − − ¯− we find that the real part of Z| and Z| cancels the Goldstone Fig. 1 The scalar potential V = μ 2e AA 2J V as a function of Re(A) and Im(A) at J = 0 mode of (15). Similarly, applying the derivatives Dα˙ and Dα to (11) and taking their lowest components (recalling then Dα˙ Dα V |=σ m B ), we conclude that the vector field “eats 1 αα˙ m W=μ(α + β) = μ a + b − δ , (22) up” the Goldstone mode indeed, as 2 √ √ = + ∂ ζ. Bm Bm m (18) where a ≡ ( 3 − 1) and b ≡ (2 − 3) are the SUSY breaking vacuum solutions to the Polonyi parameters in the 4 Adding a cosmological constant absence of a cosmological constant (Sect. 2).

A cosmological constant (or dark energy) can be introduced into our framework without breaking any symmetries, via a 5 Scalar potential and vacuum stability simple modification of the Polonyi sector and its parameters α and β introduced in Sect. 2.3 For completeness, the stability of our vacuum solutions Just adding a (very) small positive constant δ and assuming should also be examined. On the one hand, in our model the = vacuum stability is almost guaranteed because both functions that J 0 at the minimum of the potential modify the 2 (Minkowski) vacuum condition V = 0 of Sect. 2 to J and J enter the scalar potential = 1 2 + μ2 AA¯+2J 2 α2 2 V J e V = μ e δ = m / δ. (19) 2 3 2 2 ¯ 2 J 2 By comparing the condition (19)toEq.(6) we find the rela- × |1 + Aβ + AA| − 3 − 2 |A + β| (23) J tion with the positive sign, while the function J is required to 2 2 2 (1 + αβ + α ) − 3(α + β) = δ. (20) be positive for the ghost-freedom. On the other hand, the only term with the negative sign in the scalar potential (23) = 2 δ A solution to Eqs. (20) and (7) with V m3/2 is the true is −3|A + β|2 but it grows slower than the positive quartic minimum, and it reads | + β + ¯|4 √ term 1 A AA . √ − The non-negativity of the scalar potential (23)for|A| < 1 α = ( − ) + 3√ 2 3 δ + O(δ2), 3 1 is not as apparent as that for |A|≥1. That is why we supply 3( 3 − 1) √ (21) Figs. 1 and 2 where the non-negativity becomes apparent √ 3 − 3 β = (2 − 3) + √ δ + O(δ2). too. In accordance to the previous Sect. 4, we can also add 6( 3 − 1) a positive cosmological constant that shifts the minimum to = 2 δ This yields a de Sitter vacuum with the spontaneously broken V m3/2 describing a de Sitter vacuum. SUSY after inflation. Inserting the solution into the superpotential and ignoring the O(δ2)-terms, we find 6 Conclusion

3 A similar idea was used in Ref. [13], though in the different context, Our new results are given in Sects. 3, 4 and 5. The new gauge- where the Polonyi potential was needed to prevent the real part of the invariant formulation of our models can be used for unifica- stabilizer field from vanishing at the minimum by imposing the condition mgravitino minflaton. In our approach, there is no stabilizer tion of inflation with super-GUT in the context of supergrav- field, while the inflation comes from the D-type potential. ity, and it has a single inflaton scalar field having a positive 123 Eur. Phys. J. C (2017) 77:233 Page 5 of 6 233

0.8 to the electro-weak scale Ð see e.g., Ref. [14] for the previous studies along these lines.

0.6 Our models can be further extended in the gauge sector to the BornÐInfeld-type gauge theory coupled to supergravity and other matter, along the lines of Refs. [15,16], thus provid- 0.4 ing further support toward their possible origin in superstring (ﬂux-) compactiﬁcation.

0.2 Acknowledgements Y.A. is supported by a scholarship from the Min- istry of Education, Culture, Sports, Science and Technology (MEXT) in Japan. S.V.K. is supported by a Grant-in-Aid of the Japanese Society Re A 0.2 0.4 0.6 0.8 1.0 for Promotion of Science (JSPS) under No. 26400252, a TMU President Grant of Tokyo Metropolitan University in Japan, the World Premier Fig. 2 The real slice at Im(A) = 0ofFig.1 around the minimum of International Research Center Initiative (WPI Initiative), MEXT, Japan, V˜ and the Competitiveness Enhancement Program of Tomsk Polytechnic University in Russia. The authors are grateful to the referees for their critical comments. definite scalar potential, a spontaneous SUSY breaking and a de Sitter vacuum after inflation. Our approach does not Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm preserve the R-symmetry. ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, Our upgrade of the earlier results in Ref. [8] is not limited and reproduction in any medium, provided you give appropriate credit to the generalized matter couplings in supergravity, given to the original author(s) and the source, provide a link to the Creative by Eqs. (12) and (13). The standard approach to inflation Commons license, and indicate if changes were made. Funded by SCOAP3. in supergravity is based on the assumption that the inflaton belongs to a chiral (scalar) multiplet. It leads to the well-known problems such as the so-called η-problem, sta- bilization of other scalars, getting SUSY breaking and a dS References vacuum after inflation, etc. Though some solutions to these problems exist in the literature, they are rather complicated 1. Planck Collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. and include the additional “hand-made” input such as extra Cosmological parameters. arXiv:1502.01589 [astro-ph.CO] (stabilizing) matter superfields, extra (shift) symmetries or 2. Planck Collaboration, P.A.R. Ade et al., Planck 2015 results. XX. extra (nilpotency) conditions. We advocate another approach Constraints on inflation. arXiv:1502.02114 [astro-ph.CO] where inflaton is assumed to belong to a massive vector mul- 3. BICEP2, Keck Array Collaboration, P.A.R. Ade et al., Improved constraints on cosmology and foregrounds from BICEP2 and Keck tiplet, while SUSY breaking and a dS vacuum are achieved array cosmic microwave background data with inclusion of 95 with the help of a Polonyi superfield. It is much simpler and GHz band. Phys. Rev. Lett. 116, 031302 (2016). arXiv:1510.09217 more flexible than the standard approach. [astro-ph.CO] Physical applications of our approach to super-GUT and 4. M. Yamaguchi, Supergravity based inflation models: a review. Class. Quant. Grav. 28, 103001 (2011). arXiv:1101.2488 [astro- reheating are crucially dependent upon the way how the ph.CO] fields present in our models interact with the super-GUT 5. S.V. Ketov, Supergravity and early universe: the meeting point fields. Consistency of sGUT with inflation may lead to some of cosmology and high-energy physics. Int. J. Mod. Phys. A 28, new constraints on both. For instance, inflaton couplings to 1330021 (2013). arXiv:1201.2239 [hep-th] −3 6. S. Ferrara, R. Kallosh, A. Linde, M. Porrati, Minimal super- other matter have to be smaller than 10 , in order to pre- gravity models of inflation. Phys. Rev. D 88(8), 085038 (2013). serve flatness of the inflaton scalar potential and match the arXiv:1307.7696 [hep-th] observed spectrum of CMB density perturbations. In partic- 7. A. Van Proeyen, Massive vector multiplets in supergravity. Nucl. ular, Yukawa couplings of inflaton to right-handed (sterile) Phys. B 162, 376 (1980) 8. Y. Aldabergenov, S.V. Ketov, SUSY breaking after inflation in neutrino are crucial in addressing leptogenesis via inflaton supergravity with inflaton in a massive vector supermultiplet. Phys. decay and the subsequent reheating via decays of the right- Lett. B 761, 115 (2016). arXiv:1607.05366 [hep-th] handed neutrino into visible particles of the Standard Model. 9. J. Polonyi, Generalization of the Massive Scalar Multiplet Coupling Unfortunately, all this appears to be highly model-dependent to the Supergravity. 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13. A. Linde, On inflation, cosmological constant, and SUSY breaking. 15. H. Abe, Y. Sakamura, Y. Yamada, Massive vector multiplet infla- JCAP 1611, 002 (2016). arXiv:1608.00119 [hep-th] tion with DiracÐBornÐInfeld type action. Phys. Rev. D 91, 125042 14. J. Ellis, H.-J. He, Z.-Z. Xianyu, Higgs inflation, reheating and grav- (2015). arXiv:1505.02235 [hep-th] itino production in no-scale supersymmetric GUTs. JCAP 1808, 16. S. Aoki, Y.Yamada,More on DBI action in 4D N = 1 supergravity. 068 (2016). arXiv:1606.02202 [hep-th] arXiv:1611.08426 [hep-th]

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