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

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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 [1–3] 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 – 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 Calabi–Yau 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 compo- nents (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 superpo- tential 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 (Volkov–Akulov) 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.
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