Flipped SU(5) Predicts $\delta T/T$

We discuss hybrid inflation in supersymmetric flipped SU(5) model such that the cosmic microwave anisotropy $\delta T/T$ is essentially proportional to $(M /M_{P})^2$, where $M$ denotes the symmetry breaking scale and $M_{P}$ ($=2.4\times 10^{18}$ GeV) is the reduced Planck mass. The magnitude of $M$ determined from $\delta T/T$ measurements can be consistent with the value inferred from the evolution of SU(3) and SU(2) gauge couplings. In other words, one could state that flipped SU(5) predicts (more precisely `postdicts') $\delta T/T$. The scalar spectral index $n_s = 0.993\pm 0.007$, the scalar to tensor ratio satisfies $r \lapproxeq 10^{-6}$, while $dn_s/d{\rm ln}k \lapproxeq 4\times 10^{-4}$.

In a class of realistic supersymmetric (SUSY) models, inflation is associated with the breaking of some gauge symmetry G, such that the cosmic microwave anisotropy is essentially proportional to (M/M P ) 2 , where M (∼ M GU T = 2 − 3 × 10 16 GeV) denotes the symmetry breaking scale [1]. The simplest example of G is provided by U(1) B−L , and more complicated examples based on SU(5) [2] and SO(10) [3] have also been presented. The Higgs sector in these grand unified models is typically rather complicated, so that strictly speaking, the scale gM cannot be identified with the gauge coupling unification scale (here g denotes the gauge coupling associated with G). For instance, in the SO(10) example inflation is associated with the breaking of U(1) B−L rather than its SU(5) subgroup. In this letter, we hope to overcome this hurdle by identifying G with SU(5) × U(1) X , the so-called flipped SU(5) model [4].
This model is known to possess several advantages over standard grand unified models such as SU (5) and SO (10) itself that have often been discussed in the literature. A particularly compelling case is provided by the ease with which the doublet-triplet (D-T) splitting can be achieved in models based on G. Another potential advantage is the absence of topological defects especially monopoles that could create cosmological difficulties.
In this letter, we wish to highlight yet another advantage of models based on G, namely the ease with which a predictive hybrid inflation scenario [1] can be realized which is consistent with D-T splitting and works within a minimal Higgs framework. This is in contrast with recent attempts to construct analogous models based on SU(5) [2] and SO(10) [3], which turn out to require non-minimal Higgs sectors including gauge singlet fields. Although the symmetry breaking scale determined from δT /T in the latter case turns out to be comparable to the scale M GU T determined from the evolutions of the three low energy gauge couplings, an identification of the two scales is not quite possible, partly arising from the fact that there are extra Higgs fields, and also the fact that one has to resort to 'shifted' hybrid inflation [5] to avoid the monopole problem. These two issues it appears can be nicely evaded in the flipped model, so that one could argue that δT /T is predicted and turns out to be in excellent agreement with the observations [6]. This is the first model of inflation we are aware of in which this claim can be made. Other testable predictions include n s = 0.993 ± 0.007, dn s /dlnk < ∼ 4 × 10 −4 , and the scalar to tensor ratio r < ∼ 10 −6 . A U(1) R symmetry plays an essential role in the construction of this predictive inflationary scenario [1,7]. We will find that its presence implies a "double seesaw" mechanism for realizing suitable masses both for 'right' handed and the light neutrinos. Inflation is followed by (non-thermal) leptogenesis [8] with the reheat temperature consistent with the gravitino constraint [9].
Flipped SU(5) (= SU(5)×U(1) X ) is a maximal subgroup of SO(10), and contains sixteen chiral superfields per family: Here the subscript refers to the U(1) X charge in the unit of 1 √ 40 . 3 The MSSM hypercharge is given by a linear combination of a diagonal SU(5) generator and U(1) X charge operator; where Z = diag.(−1/3, −1/3, −1/3, 1/2, 1/2) is the MSSM hypercharge operator in the standard SU(5) model. Comparison with standard SU(5) reveals the interchanges, Since ν c has replaced e c in the 10-plet, the latter now belongs to the SU(5) singlet representation. The MSSM electroweak Higgs doublets reside in two five dimensional representations as follows: Comparing to the standard SU(5), H u and H d are replaced each other; The breaking of SU(5)×U(1) X to the MSSM gauge group is achieved by providing superlarge VEVs to Higgs of 10 dimensional representations, namely 10 H and 10 H along the ν c , ν c directions. We will provide a very simple superpotential shortly showing how this is achieved. But first let us briefly recall how the D-T splitting problem is elegantly solved in this framework. Consider the superpotential couplings, by the gauge symmetry can be avoided. Indeed, if this coefficient can be of order M W rather than M GU T , we would achieve two worthwhile goals. Namely, the MSSM µ problem would be resolved and dimension five nucleon decay would be essentially eliminated. The need for some additional symmetry is also mandated by our desire to implement a predictive inflationary scenario along the lines discussed in earlier papers [1,2,3,5,7]. We have found that a U(1) R symmetry is particularly potent in constraining the inflationary superpotential and will therefore exploit it here.
Disregarding the pure right handed neutrino sector for the moment, the superpotential responsible for breaking the SU(5) × U(1) X gauge symmetry, resolving the D-T splitting problem, and generating Dirac mass terms for the charged fermions and neutrinos is as follows: The quantum numbers of the superfields appearing in Eq. (7) are listed in Table I.
The U(1) R symmetry eliminates terms such as S 2 and S 3 from the superpotential, which yields a predictive inflationary scenario [1]. Higher dimensional baryon number violating operators such as 10 i 10 j 10 k 5 l S /M 2 P , 10 i 5 j 5 k 1 l S /M 2 P , etc. are heavily suppressed as a consequence of U(1) R . Thus, we expect proton decay to proceed via dimension six operators mediated by the superheavy gauge bosons. The dominant decay mode is p → e + /µ + , π 0 and the estimated lifetime is of order 10 36 yrs [10,11].
We note that the 'matter' superfields (and 10 H ) are neutral under U(1) R , so that the Z 2 subgroup of the latter does not play the role of 'matter' parity. An additional Z 2 symmetry ('matter' parity) has been introduced to avoid undesirable couplings such as 10 H 10 i 5 h , 10 H 5 i 5 h , 10 H 10 i 10 j 5 k S, 10 H 5 i 5 j 1 k S, etc. 4 This Z 2 ensures that the LSP is absolutely stable and consequently a desirable candidate for CDM.
The superpotential in Eq. (7) and the "D-term" potential, in the global SUSY limit, possesses a ground state in which the scalar components (labelled by the same notation as the corresponding superfield) acquire the following VEVs: 4 10 H 10 i 5 h and 10 induce superheavy mass terms of d c i , L i , and H u . 10 H 10 i 10 j 5 k S and 10 H 5 lead to proton as well as LSP decays.
Thus, the gauge symmetry SU(5) × U(1) X is broken at the scale M, while SUSY remains unbroken. In a N = 1 supergravity framework, after including the soft SUSY breaking terms, the S field acquires a VEV proportional to the gravitino mass m 3/2 [7]. As explained above, unwanted triplets in 10-plet and 5-plet Higgs become superheavy by the λ couplings in Eq. (7). From the y ij terms in Eq. (7), dtype quarks and charged leptons acquire masses after electroweak symmetry broken.
term provides u-type quarks and neutrino with (Dirac) masses. Since u c i and L i are contained in a single multiplet 5 i , the mass matrices for the u-type quarks and Dirac neutrinos are related in flipped SU (5): We point out here that U(1) R forbids the bare mass term 5 h 5 h , so that the electroweak Higgs doublets do not acquire superheavy masses. To resolve the MSSM µ problem we invoke, following [12], the following term in the Kähler potential: Intermediate scale SUSY breaking triggered by the hidden sector superfield Σ via The quantum numbers of Σ are listed in Table I. To realize the simplest inflationary scenario, the scalars must be displaced from their present day minima. Thus, for S >> M, the scalars 10 H and 10 H → 0, so that the gauge symmetry is restored but SUSY is broken. This generates a tree level scalar potential V tree = κ 2 M 4 , which will drive inflation. In practice, in addition to the supergravity corrections and the soft SUSY breaking terms, we also must include one loop radiative corrections arising from the fact that SUSY is broken by S = 0 during inflation. This causes a split between the masses of the scalar and fermionic components in 10 H , 10 H . For completeness, following Refs. [14,15], we provide here the inflationary potential that is employed to compute the CMB anisotropy δT /T , the scalar spectral index n s , and the tensor to scalar ratio r: where z ≡ |S 2 |/M 2 , N (= 10) denotes the dimensionality of 10 H , 10 H , and Λ is a renormalization mass. We have employed a minimal Kähler potential and a = 2|2 − A| cos[argS + arg(2 − A)], where A denotes the "A-parmater" associated with the soft terms. Note that during the last 60 or so e-foldings the value of the S field is well below M P , especially for κ < ∼ 10 −2 , which means that the supergravity correction proportional to |S| 4 /M 2 P is adequately suppressed (see Fig. 1). In addition, the soft term also can be safely ignored for κ > ∼ 10 −3 .
Neglecting the supergravity correction and the soft term in Eq. (11), δT /T is given by [1] δT where g 5 (≈ 0.7) denotes the SU(5) gauge coupling. 5 In Fig. 1 we plot M versus κ,   required by the constraint T r < ∼ 10 9 GeV, we find that n s = 0.993 ± 0.007. Measurement of n s to better than a percent is eagerly awaited. Fig. 4 shows that dn s /dlnk < ∼ 4 × 10 −4 . The tensor to scalar ratio r < ∼ 10 −6 (Fig. 5). to reheat temperature T r < ∼ 10 9 GeV, in order to the gravitino problem is avoided. 6 See Fig. 5 for the dependence of T r on κ. It shows that κ < ∼ 10 −2 for T r < ∼ 10 9 GeV.
We expect the decay to proceed via the production of right handed neutrinos and sneutrinos arising from the quartic (dimension five) superpotential couplings that can be produced by the decaying inflaton [19]. Furthermore, the decaying right handed neutrinos can provide a nice explanation of the observed baryon asymmetry via leptogenesis [8]. Following Ref. [15], one can derive the κ dependence of T r shown in Fig. 5. The quartic coupling above is, however, inconsistent with the U(1) R symmetry, and a somewhat more elaborate scenario based on the "double seesaw" [20] 6 Recently it was argued that the reheat temperature T r < ∼ 10 6−7 GeV for m 3/2 ∼ 100 GeV, unless the hadronic decays of the gravitino are suppressed [16]. The low temperature leptogenesis scenario (T r ∼ 10 2 GeV) could work in this class of inflationary models [15], if two right handed neutrinos with masses of 10 4 GeV or so are nearly degenerate. Note that the heaviest right handed neutrino can still have a much larger mass than the inflaton (say ∼ 10 14 GeV; thus it cannot be produced by the inflaton perturbative decay). The baryon asymmetry is then of order (T r /m χ ) × ǫ, where m χ denotes the inflaton mass and ǫ the lepton asymmetry per neutrino decay. κ ∼ 10 −5 (thus m χ ∼ 10 11 GeV) and ǫ ∼ O(1) give the desired baryon asymmetry (∼ 10 −10 ). Actually, ǫ can be as large as 1/2, provided the neutrino mass splittings are comparable to their decay widths [17]. The constraint T r < ∼ 10 9 GeV remains intact if either the hadronic decay ratio of the gravitino is small enough ( < ∼ 10 −3 ) and m 3/2 > ∼ 3 TeV, or if the gravitino happens to be the LSP. The gravitino, in principle, could have many decay channels into hidden sector fields, which would be helpful for lowering its hadronic decay ratio. Moreover, if the gravitino is the LSP and T r ∼ 10 10 GeV, the gravitino can be the dark matter in the universe [18].
is required to implement both the usual seesaw mechanism and the desired reheat scenario. The details of one such extension are as follows.

Table II
Note that the superfields appearing in Table I are neutral under U(1) H . From the "D-term" scalar potential associated with U(1) H , the scalar components of Φ and Φ ′ develop non-zero VEVs; |Φ| 2 + |Φ ′ | 2 = √ ξ ∼ 10 17 GeV, while Ψ vanishes by including the soft terms in the potential. Here the parameter ξ comes from the "Fayet-Iliopoulos D-term" [21]. Since ξ >> κ 2 M 4 /M 2 P , U(1) H is broken even during inflation. Thus, cosmic strings associated with U(1) H breaking would be inflated away.
By including soft SUSY breaking terms and supergravity effects from the higher order Kähler potential term, where h is real and −1 < ∼ h < 0, Φ and Φ ′ could be determined such that | Φ | = | Φ ′ |. We have assumed here that terms proportional to (ΦΦ † ) 2 and (Φ ′ Φ ′ † ) 2 in the Kähler potential are suppressed relative to the one given in Eq. (16). It turns out that With Φ ∼ Φ ′ ∼ 10 17 GeV, Ψ has a superheavy Majorana mass of order ρ ΦΦ ′ /M P ∼ 10 16 GeV, while Ψ and ν c i obtain (psuedo-) Dirac masses of order ρ i Φ ν c H /M P < ∼ 10 15 GeV. Hence, the "seesaw masses" (∼ [ρ 2 of the lighter mass eigenstates, which are indeed the "physical" right handed neutrinos, should be of order 10 14 GeV or smaller, as desired. In summary, it is tempting to think that inflation is somehow linked to grand unification, especially since the scale associated with the vacuum energy that drives inflation should be less than or of order 10 16 GeV. Models in which δT /T is proportional to (M/M P ) 2 , with M comparable to M GU T are especially interesting in this regard [1]. Supersymmetric flipped SU(5) provides a particularly compelling example in which the magnitude of δT /T can be 'predicted' by exploiting the gauge coupling unification scale that has been known for several years. Precise measurement of the scalar spectral index will provide an important test of this class of models.

Acknowledgments
We thank Nefer Senoguz for helpful discussions and for providing us with the figures.
Q.S. is partially supported by the DOE under contract No. DE-FG02-91ER40626.
Q.S. acknowledges the hospitality of Professors Eung Jin Chun and Chung Wook Kim at KIAS where this project was initiated. 7 Alternatively, Φ and Φ ′ could be determined by introduction of Φ with proper quantum numbers assigned and the nonrenormalizable terms in the superpotential, W ⊃ S(κ ′ ΦΦ − ρ ′ (ΦΦ) 2 /M 2 P ). By including soft terms, it turns out Φ = Φ = κ ′ M 2 P /ρ ′ at a local minimum [3]. From the "D-term" potential, we have | Φ ′ | 2 = ξ. We should assmue κ ′ /ρ ′ << 1 to keep intact the inflationary scenario discussed so far. In this case also the VEV of the lighter mass eigenstate vanishes both during and after inflation.      GeV κ Figure 6: The lower bound on the reheat temperature T r (solid) and the inflaton mass m χ (dashed) vs. κ.