Gravity Waves and Proton Decay in a Flipped SU(5) Hybrid Inflation Model

PHYSICAL REVIEW D 97, 123522 (2018)

Gravity waves and proton decay in a flipped SU(5) hybrid inflation model

† ‡ Mansoor Ur Rehman,1,* Qaisar Shafi,2, and Umer Zubair2, 1Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan 2Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA

(Received 10 April 2018; published 14 June 2018)

We revisit supersymmetric hybrid inflation in the context of the flipped SUð5Þ model. With minimal superpotential and minimal Kähler potential, and soft supersymmetry (SUSY) masses of order (1–100) TeV, compatibility with the Planck data yields a symmetry breaking scale M of flipped SUð5Þ close to ð2–4Þ × 1015 GeV. This disagrees with the lower limit M ≳ 7 × 1015 GeV set from proton decay searches by the Super-Kamiokande collaboration. We show how M close to the unification scale 2 × 1016 GeV can be reconciled with SUSY hybrid inflation by employing a nonminimal Kähler potential. Proton decays into eþπ0 with an estimated lifetime of order 1036 years. The tensor to scalar ratio r in this case can approach observable values ∼10−4–10−3.

DOI: 10.1103/PhysRevD.97.123522

I. INTRODUCTION flipped SUð5Þ gauge group see [9] where each of two hybrid fields is shown to realize inflation. For no-scale The supersymmetric (SUSY) hybrid inflation model 5 – SUSY flipped SUð Þ models of inflation see [10,11]. [1 5] has attracted a fair amount of attention due to its 5 ≡ 5 1 The flipped SUð Þ SUð Þ × Uð ÞX model [12,13] simplicity and elegance in realizing the grand unified exhibits many remarkable features and constitutes an attrac- theory (GUT) models of inflation [5]. In models with tive choice as a grand unified gauge group. In the flipped minimal Kähler potential, the soft linear and mass squared SUð5Þ model, the doublet-triplet splitting problem is terms play an important role in attaining the scalar spectral elegantly solved due to the missing partner mechanism index compatible with the current experimental observa- [13]. The proton decay occurs via dimension-6 operators tions [6,7]. The next important task is to explore the and is naturally long lived with M around the GUT scale. possibility of realizing the gauge symmetry breaking scale Moreover, it lacks the monopole problem that appears in the M close to a typical GUT scale ∼2 × 1016 GeV. This can, spontaneous breaking of other GUT gauge groups [i.e., 5 4 2 2 10 in turn, adequately suppress the proton decay rate from SUð Þ, SUð ÞC × SUð ÞL × SUð ÞR or SOð Þ]. This dimension-6 operators usually present in GUT models. property also makes the flipped SUð5Þ model an appropriate Achieving M ∼ 2 × 1016 GeV was one of the main pre- choice for the standard version of SUSY hybrid inflation dictions of the original SUSY hybrid inflation model where where gauge symmetry is broken after the end of inflation. 5 only radiative correction was included in otherwise a flat Finally, flipped SUð Þ is also regarded as a natural GUT potential [1]. We, therefore, investigate the possibility of model due to its connection with F-theory [14]. realizing large enough M in the SUSY hybrid inflation It is important to note that the phrase gravity waves in the title refers to potentially observable primordial gravitational model with minimal Kähler potential, including various waves. The prediction of primordial gravitational waves is important corrections [1,3,5–7]. Specifically, we update the a generic feature of the inflation paradigm and originates status of the SUSY flipped SUð5Þ hybrid inflation model from the quantum nature of gravity. These gravity waves [7,8] with minimal Kähler potential and soft SUSY masses ∼1–100 are expected to be observed indirectly through the detection TeV. For other hybrid models of inflation in of B-mode polarization data in the cosmic microwave background anisotropies. Their detection determines the *[email protected] value of the tensor to scalar ratio r, which is usually † [email protected] predicted in a wide range by the various inflation models. ‡ [email protected] The main goal of future experiments [15,16] is therefore the measurement of r within an uncertainty of δr ¼ 0.001. This Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. then defines the observable range of r. Therefore, we also Further distribution of this work must maintain attribution to explore the possibility of realizing this large r range in our the author(s) and the published article’s title, journal citation, model along with the gauge symmetry breaking scale M and DOI. Funded by SCOAP3. around 2 × 1016 GeV.

2470-0010=2018=97(12)=123522(9) 123522-1 Published by the American Physical Society REHMAN, SHAFI, and ZUBAIR PHYS. REV. D 97, 123522 (2018)

The outline of the paper is as follows. In Sec. II we where the scalar component of the gauge singlet superfield briefly introduce the SUSY hybrid model of flipped SUð5Þ S acts as the inflaton. The first line in Eq. (4) is relevant for that was first proposed in [8]. We update the status of this inflation and is also responsible for the gauge symmetry model with minimal Kähler potential in Sec. III and check breaking of FSUð5Þ into the MSSM as the 10-plet Higgs its compatibility with the proton lifetime constraint. The pair attains nonzero vacuum expectation value in the c ¯ c minimal model with ∼1–100 TeV scale soft SUSY masses NH; NH direction, is shown to predict fast proton decay. However, with the 10 10¯ c ¯ c 2 help of leading order nonminimal terms in the Kähler h H Hi¼hNHNHi¼M : ð5Þ potential we overcome this problem and the predictions of inflationary parameters are found to be in accordance with The second line in Eq. (4) contains the terms that are the latest Planck data. This is discussed in detail in Sec. IV. involved in the solution of the doublet-triplet splitting → þπ0 The dominant proton decay mode is p e with a problem. The Uð1Þ symmetry plays a key role here. 36 R lifetime estimated to be of order 10 years. Finally, we This symmetry not only eliminates the S2 and S3 terms to provide a brief summary of our findings in Sec. V. realize successful inflation; it also forbids the bilinear term ¯ 5h5h to avoid GUT scale masses of the MSSM Higgs II. SUSY FSU(5) HYBRID INFLATION doublets Hu and Hd. The MSSM μ problem is assumed to be solved by the Giudice-Masiero mechanism [17]. Finally, The minimal Higgs sector of flipped SUð5Þ ≡ the terms in the second line of Eq. (4) mix the color triplets FSUð5Þ ≡ SUð5Þ × Uð1Þ consists of a pair of Higgs X (Dc ; D¯ c ) and (D , D¯ ) to attain GUT scale masses. This superfields (10 ; 10¯ ), and a second pair of 5-plet Higgs H H h h H H then solves the doublet-triplet problem and eliminates superfields (5 , 5¯ ), which are decomposed under the h h dimension-5 proton decay mediated by colored Higgsino standard model (SM) gauge group as exchange. 10 10 1 3 2 1 6 c 3¯ 1 1 3 The terms in the third line of Eq. (4) generate the Dirac H ¼ð ; Þ¼QHð ; ; = ÞþDHð ; ; = Þ ν mass terms for all fermions, where yðdÞ, yðu; Þ, and yðeÞ Nc 1; 1; 0 ; ij ij ij þ Hð Þ denote the corresponding Yukawa couplings. For a dis- 10¯ 10¯ −1 ¯ 3¯ 2 −1 6 ¯ c 3 1 −1 3 H ¼ð ; Þ¼QHð ; ; = ÞþDHð ; ; = Þ cussion of light neutrino masses in this model see [8]. Another possibility to realize light neutrino masses by þ N¯ c ð1; 1; 0Þ; H assuming R breaking at nonrenormalizable level is dis- 5h ¼ð5; −2Þ¼Dhð3; 1; −1=3ÞþHdð1; 2; −1=2Þ; cussed in [18]. As all matter superfields are neutral under ¯ ¯ U 1 symmetry, an additional Z2 symmetry (or matter 5 ¼ð5; 2Þ¼D¯ ð3¯; 1; 1=3ÞþH ð1; 2; 1=2Þ: ð1Þ ð ÞR h h u parity) is assumed [8]. This symmetry not only realizes the The minimal supersymmetric standard model (MSSM) possibility of lightest supersymmetric particle (LSP) as a matter content and the right-handed neutrino reside in cold dark matter candidate but also avoids some unwanted the following representations: terms in the superpotential. In the D-flat direction, the relevant part of the global 10F 10 1 3 2 1 6 c 3¯ 1 1 3 SUSY potential may be written as i ¼ð ; Þi ¼ Qið ; ; = ÞþDi ð ; ; = Þ c 1 1 0 þ Ni ð ; ; Þ; 2 2 2 2 2 2 2 V ¼ κ ðj10Hj − M Þ þ 2κ jSj j10Hj : ð6Þ 5¯f 5¯ −3 c 3¯ 1 −2 3 1 2 −1 2 i ¼ð ; Þi ¼ Ui ð ; ; = ÞþLið ; ; = Þ; Along the inflationary valley (j10 j¼j10¯ j¼0), 1¯e ¼ð1; 5Þ ¼ Ecð1; 1; þ1Þ; ð2Þ H H i i i SUSY is temporarily broken by the vacuum energy κ2 4 where Nc is the right-handed neutrino superfield. density V0 ¼ M , and is restored later at the global 10 10¯ 0 Assuming the following R-charge assignment of the super- minimum (jh Hij ¼ jh Hij ¼ M, jhSij ¼ ). In the infla- fields, tionary trajectory, the effective contributions of one-loop radiative correction and soft SUSY breaking terms can be ¯ ¯ ¯ written as ðS;10H;10H;5h;5h;10i;5i;1iÞ¼ð1;0;0;1;1;0;0;0Þ; ð3Þ

4 the superpotential of the model is given by [8] ðκMÞ N ΔV − ≃ F x ; one loop 8π2 ð Þ ð7Þ ¯ 2 W ¼ κS½10H10H − M ¯ ¯ ¯ Δ ≃ κ 3 2 2 2 þ λ110H10H5h þ λ210H10H5h VSoft am3=2 M x þ MSM x ; ð8Þ ðdÞ10F10F5 ðu;νÞ10F5¯f5¯ ðeÞ1e5¯f5 þ yij i j h þ yij i j h þ yij i j h; ð4Þ with

123522-2 GRAVITY WAVES AND PROTON DECAY IN A FLIPPED … PHYS. REV. D 97, 123522 (2018) 1 ðx4 − 1Þ x2 þ 1 The leading order slow-roll parameters are defined as FðxÞ¼ ðx4 þ 1Þ ln þ 2x2 ln 4 x4 x2 − 1 1 m 2 V0 2 κ2M2x2 ϵ ¼ P ; þ 2 ln − 3 ð9Þ 4 M V Q2 1 m 2 V00 η ¼ P ; 2 M V and 1 m 4 V0V000 ξ2 ¼ P ; ð16Þ 4 M V2 a ¼ 2j2 − Aj cos½arg S þ argð2 − AÞ: ð10Þ 18 where mP ¼ 2.4 × 10 GeV is the reduced Planck mass. Here, N ¼ 10 is the dimensionality of the 10-plet Higgs In the leading order slow-roll approximation, the scalar conjugate pair, Q is the renormalization scale and we have spectral index ns, the tensor-to-scalar ratio r and the running of the scalar spectral index dns=d ln k are given by defined x ≡ jSj=M. The a and MS are the coefficients of soft SUSY breaking linear and mass terms for S, respec- ns ≃ 1 þ 2η − 6ϵ; ð17Þ tively, and m3=2 is the gravitino mass.

r ≃ 16ϵ; ð18Þ III. MINIMAL KÄHLER POTENTIAL dn In order to include the supergravity (SUGRA) correction s ≃ 16ϵη − 24ϵ2 − 2ξ2: ð19Þ we first consider the minimal canonical Kähler potential, d ln k

For negligibly small values of r and dns , the relevant 2 2 ¯ 2 d ln k K ¼jSj þj10Hj þj10Hj : ð11Þ Planck constraint on the scalar spectral index ns in the base ΛCDM model is [19] The F-term SUGRA scalar potential is given by ns ¼ 0.9677 0.0060

K=m2 −1 −2 2 ð68%CL; PlanckTT þ lowP þ lensingÞ: ð20Þ V ¼ e P ðK ¯ D WD W − 3m jWj Þ; ð12Þ SUGRA ij zi zj P The amplitude of the primordial spectrum is given by with zi being the bosonic components of the superfields ¯ 1 4 zi ∈ fS; 10H; 10H; g, and we have defined V=mP Asðk0Þ¼24π2 ϵ ; ð21Þ x¼x0 ∂ ∂ ∂2 W −2 K K −9 D W ≡ þ m W; K ¯ ≡ ; ð13Þ and has been measured by Planck to be A ¼ 2.137 × 10 zi ∂ P ∂ ij ∂ ∂ s zi zi zi zj −1 at k0 ¼ 0.05 Mpc [19]. The last N0 number of e-folds before the end of inflation is and D W ¼ðD WÞ . Putting all these corrections zi zi Z 2 x0 together, we obtain the following form of inflationary 2 M V N0 ¼ 0 dx; ð22Þ potential, mP xe V

where x0 is the field value at the pivot scale k0, and xe is the V ≃ V þ ΔV − þ ΔV ; ð14Þ SUGRA one loop Soft field value at the end of inflation. The value of xe is fixed either by the breakdown of the slow-roll approximation, or by a “waterfall” destabilization occurring at the value 4 4 κ2N 2 4 M x x ¼ 1 if the slow-roll approximation holds. ≃ κ M 1 þ þ FðxÞ c m 2 8π2 The results of our numerical calculations are depicted in P 2 Figs. 1 and 2. Following [7], we have taken a ¼ −1 m3=2x M x þ a þ S : ð15Þ assuming appropriate initial condition for arg S [20].In κM κM addition, we set the number of e-folds N0 ¼ 50 and the scalar spectral index ns is fixed at the central value (0.968) The prediction of various inflationary parameters can of Planck data bounds. The left panel of Fig. 1 shows the now be estimated using standard slow-roll definitions behavior of κ with respect to MS, while the behavior of described below. GUT symmetry breaking scale M with respect to MS is

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FIG. 1. The symmetry breaking scale M (right panel) and κ (left panel) versus soft SUSY breaking mass MS for a ¼ −1, N0 ¼ 50, and ns ¼ 0.968 (central value). The green, brown, and red curves respectively correspond to m3=2 ¼ 1, 10, and 100 TeV. The solid curves are 2 0 2 0 drawn for MS < , while the dashed curves are drawn for MS > .

shown in the right panel and is of particular importance 2 2 00 2 m κ N jF ðx0Þj 2M because of proton decay considerations. The curves are n ≃ 1 þ p − þ S : ð24Þ s M 8π2 κ2M2 drawn for different values of m3=2. The solid curves are 2 0 drawn for MS < , while the dashed curves are drawn for Next we estimate the values of κ, M, and M such that the 2 0 S MS > . To make our discussion relevant for the current contributions of the soft linear and mass squared terms are ∼1–100 experiments we restrict the soft masses to be TeV. comparable in Eq. (23). Assuming x0 ∼ 1 we obtain the This automatically includes the split-SUSY scenario where following expressions using the above equations: soft scalar masses can take values up to 100 TeV [21].In 11 6 1 8 1 4 our region of interest, the radiative correction provides the 2 π 1 − = m3 2 = κ ≃ ð nsÞ = dominant contribution while the SUGRA correction is 3 2 00 ; ð25Þ N log ð4ÞjF ðx0Þj mp mostly negligible. The suppression of the SUGRA correc- −3 3 1 8 tion is supported by the tiny values of S0=mp ≲ 2 × 10 as N 00 = jF ðx0Þj 3 1=4 M ≃ ðm3 2m Þ ; ð26Þ shown in the left panel of Fig. 2. This approximation 2π2 log2ð4Þð1 − n Þ3 = p simplifies the expressions of the amplitude of curvature s rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi perturbation and the scalar spectral index as κ2 N 4 ≃ M logð Þ jMSj 4π 2 : ð27Þ κ2M6 A ≃ ; 23 1 00 ∼−4 5 s κ2N 0 2 2 2 ð Þ For m3=2 ¼ TeV with F ðx0Þ . we obtain 2 6 F ðx0Þ MSx0 m3=2 6π mpð 2 þ 2 2 − Þ −4 15 4 8π κ M kM κ∼1.8×10 , M∼1.8×10 GeV and MS ∼1.1×10 TeV.

FIG. 2. The tensor to scalar ratio r (right panel) and S0=mP (left panel) versus soft SUSY breaking mass MS for a ¼ −1, N0 ¼ 50, and ns ¼ 0.968 (central value). The green, brown, and red curves respectively correspond to m3=2 ¼ 1, 10, and 100 TeV. The solid curves are 2 0 2 0 drawn for MS < , while the dashed curves are drawn for MS > .

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00 Similarly, for m3 2 ¼100TeV with F ðx0Þ∼−1.7 we obtain 10F 5¯f = superpotential [Eq. (4)]. The matter superfields i , j , and −4 15 4 κ∼6.5×10 , M∼4.1×10 GeV and MS ∼2.2×10 TeV. 1¯e l are odd under this Z2 matter parity whereas all other These estimates are in good agreement with the numerical superfields are even. This matter parity not only forbids data shown in Figs. 1 and 2. Therefore, with both m3=2 and many unwanted couplings as mentioned in [8], but it also MS less than 100 TeV, only the radiative correction and the makes the neutral LSP a suitable dark matter candidate. linear soft term dominate while the soft mass squared term Therefore, proton decay occurs via a dimension-6 operator and SUGRA corrections are negligibly small. For from the superheavy gauge boson exchange, and the lifetime ≳ 104 þ 0 MS TeV, the soft mass squared term begins to take for the channel p → e π is given by [25–27] 2 0 over, which drives the curve upward for MS < , and 2 4 4 0 M5 1.005 downward for MS > . Furthermore, the tensor to scalar τ → þπ0 ≈ 1035 ðp e Þ 16 × × years; ratio r turns out to be extremely small, taking on values 10 GeV g5 r ∼ 1.5 × 10−13–6.3 × 10−11. This can be seen in the right ð31Þ panel of Fig. 2 where r is plotted against MS for different values of m3=2. Our findings in this section are compatible 3 2 where M5 ¼ g5M is the SUð Þc × SUð ÞL unification scale with the results of [22] where SUSY hybrid inflation with with unified gauge coupling g5 evaluated at this scale. Note minimal Kähler and soft SUSY masses of order that the scale M5 usually lies below the unification scale of ∼ 0 1–10 1 ð . Þ TeV is considered for the Uð ÞB−L gauge group. 5 1 the SUð Þ × Uð ÞX group where g5, the gauge coupling of Finally, it is important to note that with a minimal Kähler SUð5Þ,isunifiedwithg , the gauge coupling of Uð1Þ .The ∼ 1–100 X X potential and soft SUSY masses of order ð Þ TeV, the Super-Kamiokande experiment places a lower bound on symmetry breaking scale turns out to be relatively small, proton lifetime of 1.6 × 1034 years at 90% confidence level ∼ 1 7–3 8 1015 M ð . . Þ × GeV. This leads to a fast proton decay for the channel p → eþπ0 [28,29]. This then translates into a rate as briefly discussed below. lower bound on M,

A. Proton decay in FSU(5) M>6.3 × 1015 GeV; ð32Þ In the flipped SUð5Þ model, the dangerous dimension-5 15 proton decay operators are highly suppressed due to which disagrees with the result M ∼ ð1.7–3.8Þ × 10 GeV R symmetry. For example, even though the operators stated above. Moreover, the successful breaking of the f f f 5 ∼ 1 6 10F10F10F5¯ ⊃ Q Q Q L and 10F5¯ 5¯ 1¯e ⊃ DcUcUcEc FSUð Þ model into the MSSM requires M . × i j k l i j k l i j k l i j k l 1016 ∼ 1016 ∼ 0 7 are FSUð5Þ gauge invariant, they are not invariant under GeV for M5 GeV and g5 . . These issues R symmetry. Further, consider the following R-symmetric can be resolved by employing a nonminimal Kähler potential FSUð5Þ gauge invariant operators [8]: in which case the symmetry breaking scale M can be raised to the desired value. ¯ f S10F10F10F5 Q Q Q L i j k l ⊃ hSi i j k l 2 ; IV. NONMINIMAL KÄHLER POTENTIAL MP MP MP S10F5¯f5¯f1¯e DcUcUcEc The nonminimal Kähler potential may be expanded as i j k l ⊃ hSi i j k l 2 : ð28Þ MP MP MP κ 4 κ 10 4 κ 10¯ 4 2 10 2 10¯ 2 S jSj H j Hj H¯ j Hj K ¼jSj þj Hj þj Hj þ 4 2 þ 4 2 þ 4 2 With softly broken SUSY, the superfield S is expected to mP mP mP 2 2 2 2 2 2 attain nonzero vacuum expectation value hSi ≃−m3 2=κ 10 10¯ 10 10¯ = κ jSj j Hj κ jSj j Hj κ j Hj j Hj [23,24]. This makes the above operators heavily sup- þ SH 2 þ SH¯ 2 þ HH¯ 2 mP mP mP pressed. There are additional R-symmetric FSUð5Þ gauge κ 6 invariant operators that lead to proton decay. These include SS jSj þ 6 4 þ: ð33Þ mP ¯f S10 10F10F5 S Nc H i j k ⊃ h i h Hi c c c c 2 ðQiDj Lk þ Di Dj UkÞ; ð29Þ Using Eqs. (4), (12), and (33) along with the radiative MP MP MP correction in Eq. (7) and soft mass terms in Eq. (8),we obtain the following inflationary potential, 10 5¯f5¯f1¯e c S H hSi hN i i j k ⊃ H ðL L EcÞ: ð30Þ M2 M M i j k M 2 M 4 x4 κ2N P P P V ≃ κ2M4 1 − κ x2 þ γ þ FðxÞ S m S m 2 8π2 P P Although heavily suppressed to have any observable 2 m3 2x signature for proton decay, these operators are not allowed = MSx þ a κ þ κ ; ð34Þ due to the additional Z2 matter parity imposed on the M M

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FIG. 3. κ (left panel) and tensor to scalar ratio r (right panel) with respect to the nonminimal coupling κS for N0 ¼ 50 and GUT symmetry breaking scale M ∼ 2 × 1016 GeV. The lighter shaded region represents the Planck 2-σ bounds, while the darker region represents the Planck 1-σ bounds. The upper and lower curves correspond to the jS0j¼mP and κSS ¼ 1 constraint, respectively. The brown shaded region represents jS0j ≥ 0.5mP.

7κ γ 1 − S 2κ2 − 3κ κ where S ¼ 2 þ S SS, and we have retained Figure 3 depicts the behavior of and the tensor to scalar 4 κ terms up to OððjSj=mPÞ Þ from SUGRA corrections (recall ratio r with respect to the nonminimal coupling S. These ¯ that during inflation both j10Hj and j10Hj are 0). We further plots resemble one another and their behavior can be assume soft masses am3=2 and MS to be ∼1–100 TeV, with understood from the following approximate relation 2 between r and κ [from Eqs. (18) and (21)]: a and MS being either positive or negative. The parameter space consistent with the Planck data 2κ2 M 4 bounds is enlarged with the addition of two nonminimal r ≃ : 3π2 ð36Þ parameters κS and κSS. However, to make our discussion Asðk0Þ mP interesting for near future experiments [15,16], we restrict ourselves to the parameter region with the largest possible This relation shows that larger values of r are expected when values of the tensor to scalar ratio r with M ∼ 2 × 1016 GeV. κ or M is large. Since M is fixed, larger r values should occur 16 As previously discussed in [30], the possibility of larger r for larger κ values. For fixed M ∼ 2 × 10 GeV, the highest −3 solutions restricts the nonminimal parameters, namely value of r (∼5 × 10 ) obtained in our numerical results κ ≃ 0 2 κS < 0 and κSS > 0 with the quartic coupling γS < 0. occurs for . (see Fig. 3). In the leading order slow-roll Therefore, large r solutions are obtained mainly with a approximation, the spectral index ns and tensor to scalar ratio potential of the form r are given by 2 2 N κ2 00 V ≃ 1 − 2κ 6γ M 2 mP F ðx0Þ ⊃ 1 þ Quadratic-Quartic: ð35Þ ns S þ S x0 þ 2 ; V0 mP M 8π ð37Þ The linear and soft mass squared terms with am3 2 and M ∼ = S 1–100 TeV are suppressed, while the radiative and SUGRA 2 4 2 N κ2 0 2 ≃4 mP 2γ M 3 −2κ M F ðx0Þ corrections parametrized by κ and κ play the dominant r S x0 S x0 þ 2 : S SS M m m 8π role. To keep the SUGRA expansion under control we further P P limit S0 ≤ mP. We also require that the nonminimal cou- ð38Þ plings jκ j ≤ 1 and jκ j ≤ 1. Using next to leading order S SS ≃ slow-roll approximation, the results of our numerical calcu- Solving these two equations simultaneously for S0 mP, −3 lations are displayed in Figs. 3–6. The lighter (darker) region r ≃ 5 × 10 ,andns ≃ 0.968 (central value) we obtain κS ≃ −0 03 γ ≃−0 014 ∼ 0 5 represents the Planck 2-σ (1-σ) bounds on r and ns.The . and S . . Further, with jS0j . mP and ∼ 10−3 κ ∼ 0 09 κ ≃−0 03 γ ≃−0 06 upper and lower cutoffs correspond to the S0 ¼ mP and r ,weget . , S . , and S . . κSS ¼ 1 constraints. We have also included a brown shaded These estimates are in good agreement with our numerical – κ γ region for jS0j ≳ 0.5mP. This is the ultraviolet sensitivity results displayed in Figs. 3 6. The behavior of SS and S region where higher order Planck suppressed terms in the with respect to κS is presented in Fig. 4, while Fig. 5 depicts SUGRA expansion become important. Outside this region the behavior of S0=mP with respect to κS and γS. with jS0j ≲ 0.5mP we not only obtain a natural suppression To facilitate this discussion further, we have also of higher order terms but also ensure the boundness of the provided a plot of r versus γS in the left panel of Fig. 6 −3 potential, a problem arising due to γS < 0 [31]. where it can be seen that larger values of rð∼10 Þ are

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FIG. 4. κSS (left panel) and γS (right panel) with respect to the nonminimal coupling κS for N0 ¼ 50 and GUT symmetry breaking scale M ∼ 2 × 1016 GeV. The lighter shaded region represents the Planck 2-σ bounds, while the darker region represents the Planck 1-σ bounds. The upper and lower curves correspond to the κSS ¼ 1 and jS0j¼mP constraints, respectively. The brown shaded region represents jS0j ≥ 0.5mP.

FIG. 5. jS0j=mP versus nonminimal coupling κS (left panel) and γS (right panel) for N0 ¼ 50 and GUT symmetry breaking scale M ∼ 2 × 1016 GeV. The lighter shaded region represents the Planck 2-σ bounds, while the darker region represents the Planck 1-σ bounds. The upper and lower curves correspond to the jS0j¼mP and κSS ¼ 1 constraints, respectively. The brown shaded region represents jS0j ≥ 0.5mP. The brown shaded region represents jS0j ≥ 0.5mP.

FIG. 6. Tensor to scalar ratio r versus the quartic coupling γS (left panel) and the running of scalar spectral index dns=d ln k (right 50 1 ∼ 1 2 0 ∼ 2 1016 panel) for N0 ¼ , a ¼ , m3=2 ¼ MS TeV (MS > ) and GUT symmetry breaking scale M × GeV. The lighter shaded region represents the Planck 2-σ bounds, while the darker region represents the Planck 1-σ bounds. The upper and lower curves correspond to the jS0j¼mP and κSS ¼ 1 constraints, respectively. The brown shaded region represents jS0j ≥ 0.5mP.

123522-7 REHMAN, SHAFI, and ZUBAIR PHYS. REV. D 97, 123522 (2018) obtained for smaller values of γSð∼−0.06Þ. It is important V. SUMMARY to note that large tensor modes can be obtained for any To summarize, we have revisited supersymmetric hybrid value of scalar spectral index n within Planck bounds. In s inflation in the framework of the flipped SUð5Þ model. We short, for nonminimal couplings −0.034 ≤ κS ≤−0.027 0 37 ≤ κ ≤ 1 have shown that with a minimal Kähler potential and soft and . SS , we obtain the scalar spectral index ns – 2 σ SUSY masses of order (1 100) TeV, this model predicts a within the Planck - bounds and tensor to scalar ratio r in ∼ 2–4 1015 3 10−5–5 10−3 symmetry breaking scale M ð Þ × GeV, for the the range ( × × ). Moreover, with the sym- central value n 0.968. This value of M is significantly ∼2 1016 s ¼ metry breaking scale fixed at ( × GeV), the proton below the GUT unification scale 2 × 1016 GeV and leads to ∼2 1036 is naturally stable with a lifetime of × years. proton lifetime τ ∼ 1032–33 years as compared to the current The right panel of Fig. 6 shows the dependence of the 34 lower limit τ þ 0 ≳ 1.6 × 10 years determined by the spectral running dn =d ln k on tensor to scalar ratio r. It can p→e π s Super-Kamiokande collaboration. The tensor to scalar ratio be seen that the spectral running does not vary appreciably also turns out to be extremely small, taking on values with r and takes on roughly the same values for large and r ∼ ð10−13–7 × 10−11Þ. By employing nonminimal Kähler small r values. The scalar spectral running dns=d ln k varies – – potential, with soft SUSY masses of order 1 100 TeV, the in the range (0.0024 0.0034) and this justifies the use of the ∼ 2 1016 latest Planck data of ΛCDM+tensors with no running for symmetry breaking scale M × GeV is easily achieved within the Planck σ bounds on ns and the proton the purpose of comparing the predictions of this model. 36 −5 is naturally of order ∼10 years. Moreover, larger tensor Finally, smaller r values (∼10 ) are obtained for S0 ≲ ∼10−4–10−3 0.1m and large κ ≃ 1 for which γ is negative and rather modes with observable values ( ) are obtained P SS S −0 034 ≤ κ ≤−0 027 large (∼−2), as depicted in Figs. 4–6. Since both the with nonminimal couplings . S . and 0.37 ≤ κSS ≤ 1. quadratic and quartic couplings (κS, γS) are negative in this region, the form of potential remains the same as in ACKNOWLEDGMENTS Eq. (35).ForκS > 0 and γS < 0 with soft masses ∼1–100 TeV, only tiny values of r ≲ 10−12 are obtained This work is partially supported by DOE Award No. DE- as discussed recently in [32]. SC0013880 (Q. S.).

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