A class of finite GUTs in curved spacetime is considered in connection with the cosmological inflation scenario. It is confirmed that the use of the running scalar-gravitational coupling constant in these models helps realizing a successful chaotic inflation. The analyses are made for some different sets of the models

Mukaigawa, S.

Muta, Taizo

Odintsov, S.D.

International Journal of Modern Physics A

Hiroshima University Institutional Repository

arXiv:hep-ph/9709299 v4   13 Sep 1997HUPD-9716Sept 13, 1997Finite Grand Unified Theories and InflationS. Mukaigawa, T. Muta and S. D. Odintsov †Department of Physics, Hiroshima UniversityHigashi-Hiroshima, Hiroshima 739AbstractA class of finite GUTs in curved spacetime is considered in con-nection with the cosmological inflation scenario. It is confirmed thatthe use of the running scalar-gravitational coupling constant in thesemodels helps realizing a successful chaotic inflation. The analyses aremade for some different sets of the models.1 IntroductionIt is now a common understanding to assume the presence of the inflationarystage in the early universe (see Ref. 1 for a review). Among various mod-els in the inflationary scenario the chaotic inflation model seems to be themost successful and promising.1 In the chaotic inflation model, however, weneed the fine-tuning of some coupling constants such as scalar-gravitationalcoupling constant2 ξ. The scalar-gravitational term associated with this cou-pling constant is required in any quantum field theory in curved space-timein order to guarantee the multiplicative renormalizability of the theory3 (seeRef. 4 for a general review). Applying the renormalization group argumentwe find that the coupling constant ξ starts running.3,4,5 The behavior of therunning coupling constant at strong gravitational field has been investigatedfor various models in Refs. 3 and 5 (see Ref. 4 for a review).In a recent paper6 an interesting observation has been made on an im-plication of the running coupling constant in realizing the inflation scenario.†On leave of absence from Department of Physics, Tomsk Pedagogical Institute, 634041Tomsk, Russia1The authors of Ref. 6 report that the use of the running scalar-gravitationalcoupling constant in a specific field theory7 helps constructing a successfulmodel of the chaotic inflation. The behaviour of the running ξ in their modelis typical of λϕ4-theory 3 so thatξ(t) =16+ (ξ − 16)(1− a2λt)α,λ(t) =λ1− a2λt, (1)where a2 and α are suitable constants, and t is the RG parameter, t = 12ln ϕ2µ2with µ the renormalization scale. Depending on the sign of the exponent αit is possible to have an asymptotic conformal invariance or not.3,4,5 In factξ(t) → 1/6 (for α < 0) as t → ∞ (the infrared limit ) which is the case ofRef. 6. For α > 0, |ξ(t)| → ∞ as t → ∞.3,4,5 In Ref. 6 one of the simplestsupersymmetric model, i.e. the Wess-Zumino model, was taken into accountand the effective potential in the flat direction of the model was considered inorder to restrict oneself to the effect of quadratic terms of the model. Thusthe model exhibits the typical behavior as mentioned above.There is an another type of supersymmetric models which are called finiteGUTs in which the behaviour of ξ(t) is qualitatively different from the onewhich we have seen in Eq. (1). The purpose of this note is to examine apossibility of the chaotic inflation in the finite GUTs (including some possiblefinite non-supersymmetric theories).2 Finite GUTsLet us consider the typical (supersymmetric or non-supersymmetric) finiteGUTs in curved spacetime. In curved space-time the theory is not completelyfinite8 according to an appearence of the divergence in the vacuum energy(in the external gravitational field sector). Nevertheless, the matter sectorremains unaffected.8 The behavior of coupling constants in the matter sectoris given as follows,8g2(t) = g2, h(t)2 = k1g2, f(t) = k2g2,ξ(t) =16+ (ξ − 16) exp[cg2t], (2)where g(t), h(t) and f(t) are running coupling constants respectively, g2 ≪ 1,and k1, k2 and c are certain numerical constants determined by the group2structure of the theory. Note that c may be positive, negative or zero de-pending on the nature of the theory. Let us consider a concrete example.The SU(2) gauge theory with SU(N) global invariance (the Lagrangian iswritten in the book previously mentioned;4 see Eq. (3.130) therein.) includesgauge fields, Weyl spinors and scalars in the adjoint representation of SU(2).With respect to the global group SU(N) the spinors and scalars belong tothe fundamental representation and six-dimensional antisymmetric adjointrepresentation, respectively. In the case of flat space-time the theory hasbeen introduced in Ref. 9.The theory has two regimes. In the first regime it is N = 4 extended su-persymmetric gauge theory which is finite to all orders of perturbation theoryin flat spacetime.The direct calculation yields8 c = 3/(2pi2). In the secondregime the theory corresponds to the one-loop finite non-supersymmetric the-ory in flat spacetime and c ≈ 27/(4pi)2. There are some other finite theorieswhere c could be negative or zero.8Consider the scalar sector of the finite theory taken in the flat directionof the effective potential where interaction terms do not contribute. Therenormalization-group-improved effective action in curved spacetime 4,10 cou-pled with the classical Einstein gravity is given byS = −12∫d4x√−g[Mpl8piR + Z(t)∂µϕ(t)∂µϕ(t) +m2(t)ϕ2(t)− ξ(t)Rϕ2(t)].(3)where t = 12ln ϕ2µ2. The variation of the running mass as a function of t isconsidered to be small, i.e. m2(t) ≃ m2. We choose the gauge in which theanomalous dimension for the scalar field vanishes (i.e. the renormalizationfor the scalar field is finite), thenZ(t) = 1, ϕ(t) = ϕ. (4)Under this circumstance the effective action readsS = −12∫d4x√−g[Mpl8piR + ∂µϕ∂µϕ+m2ϕ2 − ξ(t)Rϕ2]. (5)It should be noted here that in our present model it is not necessary toincorporate the effect of the running ϕ due to the property (4). In the modelemployed in Ref. 6 Eq. (4) does not hold so that the effect of the runningϕ plays an important role in deriving the cosmological predictions while inRef. 6 this effect is not fully taken into account.3Field equations for the theory characterized by Eq. (5) are given by(2−m2 + ξϕR+ 12dξdϕ(ϕ)R)ϕ = 0,Rµν − 12Rgµν =8piMplTµν . (6)Rewriting these equations in the Friedmann-Robertson-Walker Universe withscale factor a we obtain11ϕ¨+ 3Hϕ˙+m2ϕ+[ξ(ϕ)ϕ+12dξdϕ(ϕ)]×[{6ξ(ϕ)− 1 + 12 dξdϕ(ϕ)ϕ+ 6d2ξdϕ2(ϕ)ϕ2} ϕ˙2 + 2m2ϕ2 (7)+{6ξ(ϕ)ϕ+ 3 dξdϕ(ϕ)ϕ2}{ϕ¨+ 3Hϕ˙}] [Mpl8pi− ξ(ϕ)ϕ2]−1= 0,where H = a˙/a.3 Chaotic inflationAs it has been established in Refs. 11 there are two saddle points of Eq. (7)for negative ξ. They are given byϕcr = ± Mpl√−8piξ , ϕ˙ = 0. (8)It is discussed in Ref. 6 (and in the preceding works cited there) that theinitial conditions for inflaton ϕ in order to have a successful chaotic inflationas well as a sufficient period of the inflation are− Mpl√8pi|ξ|< ϕ <Mpl√8pi|ξ|, (9)and|ϕ| ≥ 5Mpl. (10)Actually for negative as well as positive ξ we have two qualitatively differentsituations summarrized by the condition (9) and (10)4Our purpose now is to examine initial conditions (9) and (10) for finiteGUTs with running ξ(t) given by Eq. (2). It should be noted that in Eq. (2)ξ = ξ(µ) and g2 = g2(µ) are initial values of the running coupling constantsξ(t) and g2(t) respectively at a certain RG scale µ . We start with a verysmall initial value of ξ(t) and so we may practically set ξ(µ) = 0. Then thetotal variation of ξ(t) comes purely from the running effect. We start withthe minimal theory at scale µ. We wish to plot ϕcr − ϕ as a function of ϕ.The relation between ϕcr − ϕ and ϕ is easily obtained by using Eq. (2) andEq. (8) with t = 12ln ϕ2µ2. In Figs.1, 2, 3 and 4 the behavior of ϕcr − ϕ isshown in four typical cases with (µ = 50Mpl, c < 0), (µ = 50Mpl, c > 0),(µ = 2Mpl, c < 0) and (µ = 2Mpl, c > 0) respectively. (Note here that thebehavior of ϕcr−ϕ as a function of ϕ is symmetric around ϕ = 0 and so weneed not to consider the case ϕ < 0).Figs. 1, 2, 3 and 4Let us examine whether we can have any region where ϕcr−ϕ is positive(i. e. ϕcr > ϕ) for small ϕ starting with ϕ ≥ 5Mpl. We clearly see that in allfour cases ϕcr − ϕ becomes positive if |c|g2 ∼ 10−3. It is important to notethat the chaotic inflation is realized independent of the sign of c. Thus weconclude that for a wide class of the finite GUTs we have successful chaoticinflations.4 ConclusionsWorking within the framework of the finite GUTs we examined the mechanism6that may lead to the successful chaotic inflation. We find that in a wide classof the finite GUTs the chaotic inflation is realized as far as the gauge couplingconstant is kept sufficiently small. In this sense the finite GUTs are worthfor further investigations in connection with the early universe scenario.In this regards it is very interesting to note that the finite GUTs in curvedspacetime are one of the possible canditates to give a solution to the cosmo-logical constant problem (see the second reference in Ref. 10) due to theexponential running of the effective cosmological constant. These faborableproperties of the theories under discussion indicate that the cosmological ap-plications of the finite GUTs (in particular the N=4 super Yang-Mills theory)should be considered more seriously.5References1. A. Linde, Particle Physics and Inflationary Cosmology, Harwood Aca-demic Publishers,1990.2. T. Futamase, T. Rothman and R. Matzner, Phys. Rev. D39 (1989)405; T. Futamase and K. Maeda, Phys.Rev. D39 (1989) 405.3. I. L. Buchbinder and S. D. Odintsov, Izw. Vuzov. Fizika (Sov. Phys.J.) N12 (1983) 108; Yad. Fiz. (Sov. J. Nucl. Phys.) 40 (1984) 1338;Lett. Nuovo Cim. 42 (1985) 379.4. I. L. Buchbinder, S. D. Odintsov and I. L. Shapiro, Effective Action inQuantum Gravity, IOP Publishing, Bristol and Philadelphia, 1992.5. T. Muta and S. D. Odintsov, Mod. Phys. Lett. A6 (1991) 3641.6. T. Futamase and M. Tanaka, preprint OCHA-PP-95, 1997; hep-ph/9704303.7. M. Tanaka, hep-th/9701063.8. I. L. Buchbinder, S. D. Odintsov and I. M. Lichtzier, Class. Quant.Grav. 6 (1989) 605.9. M.Bo¨hm and A. Denner, Nucl. Phys. B282 (1987) 206.10. I. L. Buchbinder, S. D. Odintsov, Class. Quant. Grav. 2 (1985) 721;E. Elizalde and S. D. Odintsov, Phys. Lett. B333 (1994) 331.11. L. Amendola, M. Litterio and F. Occhionero, Int. J. Mod. Phys. A5(1990) 3861; A. Barroso, J. Casasayas, P. Crawford, P. Moniz and A.Nunes, Phys.Lett. B275(1992)264.6-10-8-6-4-202460 2 4 6 8 10(ϕcr − ϕ)/Mplϕ/Mplµ/Mpl = 50 , c < 0−cg2 = 10−3−cg2 = 10−2−cg2 = 10−1Figure 1: Behavior of ϕcr − ϕ as a function of ϕ for µ = 50Mpl andc < 0. The three typical cases of the gauge coupling strength are shown:−cg2 = 10−3, 10−2, 10−1.7-10-8-6-4-202460 2 4 6 8 10(ϕcr − ϕ)/Mplϕ/Mplµ/Mpl = 50 , c > 0cg2 = 10−3cg2 = 10−2cg2 = 10−1Figure 2: Behavior of ϕcr − ϕ as a function of ϕ for µ = 50Mpl andc > 0. The three typical cases of the gauge coupling strength are shown:cg2 = 10−3, 10−2, 10−1.8-10-5051015202530350 2 4 6 8 10(ϕcr − ϕ)/Mplϕ/Mplµ/Mpl = 2 , c < 0−cg2 = 10−3−cg2 = 10−2−cg2 = 10−1Figure 3: Behavior of ϕcr − ϕ as a function of ϕ for µ = 2Mpl andc < 0. The three typical cases of the gauge coupling strength are shown:−cg2 = 10−3, 10−2, 10−1.9-10-5051015202530350 2 4 6 8 10(ϕcr − ϕ)/Mplϕ/Mplµ/Mpl = 2 , c > 0cg2 = 10−3cg2 = 10−2cg2 = 10−1Figure 4: Behavior of ϕcr − ϕ as a function of ϕ for µ = 2Mpl andc > 0. The three typical cases of the gauge coupling strength are shown:cg2 = 10−3, 10−2, 10−1.10

Finite grand unified theories and inflation

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Finite grand unified theories and inflation

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