δT /T and Neutrino Masses in SU (5)

We implement inﬂation in a supersymmetric SU (5) model with U (1) R-symmetry such that the cosmic microwave anisotropy δT/T is proportional to ( M/M Planck ) 2 , where M ∼ M GUT = 2 − 3 × 10 16 GeV, the SU (5) breaking scale, and M Planck = 2 . 4 × 10 19 GeV. The presence of a global U (1) X symmetry, spontaneously broken also at scale M GUT , provides an upper bound M 2 GUT /M Planck ∼ 10 14 GeV on the masses of SU (5) singlet right handed neutrinos, which explains the mass scale associated with atmospheric neutrino oscillations. The SU (5) monopoles and U (1) X cosmic strings are inﬂated away. Although the doublet-triplet splitting requires ﬁne-tuning, the MSSM µ problem is resolved and dimension ﬁve proton decay is strongly suppressed.

In a class of supersymmetric (SUSY) models of inflation associated with some symmetry breaking G → H, the cosmic microwave anisotropy δT /T turns out to be proportional to (M/M Planck ) 2 , where M denotes the symmetry breaking scale of G, and M Planck = 1.2 × 10 19 GeV [1,2]. Comparison with the determination of δT /T by WMAP and several other experiments [3] leads to the conclusion [1,4] that M is comparable to M GU T ∼ 2−3×10 16 GeV, the scale at which the three gauge couplings of the minimal supersymmetric standard model (MSSM) unify. In such models, the scalar spectral index of density fluctuations including supergravity (SUGRA) corrections is n s ≈ 0.98 − 1.00 [4,5], in excellent agreement with observations [3]. An essential role in the discussion is played by a global U(1) R symmetry which helps determine the form of the tree level superpotential and has other important phenomenological implications. For instance, the Z 2 subgroup of U(1) R can be identified with the "matter parity" in the MSSM. This ensures the absence of rapid proton decay and stability of the LSP. Examples of G include the groups SU(3) c × SU(2) L × SU(2) R × U(1) B−L and SU(4) c × SU(2) L × SU(2) R [6], discussed in the context of inflation in Refs. [7] and [8], respectively.
Among supersymmetric grand unified theories (GUTs) SU(5) [9] certainly is the simplest with the most direct connection to M GU T given above. It is therefore natural to try to realize this type of inflationary scenario in an SU(5) framework [10].
The minimal SU(5) model does not incorporate inflation and it also fails to provide an understanding of neutrino masses inferred from neutrino oscillations [11] and from cosmological considerations [3]. Our goal in this paper is to provide a suitable extension which can overcome both these shortcomings.
We have already indicated the important role to be played by the U(1) R symmetry. Next we consider another symmetry, namely U(1) X , which also will be important for our analysis. This is a global symmetry of the minimal model [12] and its presence prevents the appearance of tree level superheavy masses ( > ∼ M GU T ) for the SU (5) singlet right handed neutrinos. In our extended model, the breaking of U(1) X trig-gers the SU(5) breaking at M GU T . Through non-renormalizable couplings, the right handed neutrinos acquire masses of order M 2 GU T /M P ∼ 10 14 − 10 15 GeV, where M X and M P (≡ M Planck / √ 8π = 2.4 × 10 18 GeV) denote the U(1) X breaking scale and the reduced Planck mass. This can yield, via the seesaw mechanism [13], a light neutrino mass of order M 2 W /(10 14 GeV) ∼ 10 −1 eV, which is the appropriate mass scale for atmospheric neutrino oscillations. With both the global U(1) X and SU(5) spontaneously broken during inflation, the SU(5) monopoles and U(1) X cosmic strings are inflated away. At the end of inflation the oscillating fields produce right handed neutrinos whose out of equilibrium decay lead via leptogenesis to the observed baryon asymmetry of the universe [14]. The doublet-triplet splitting problem requires, as usual, some fine-tuning. However, the mechanism which generates the MSSM µ term after SUSY breaking also strongly suppresses dimension five proton decay mediated by the superheavy color triplets higgsinos.
Let us recall the well known superpotential [1,15], where κ is a dimensionless coupling constant, and the superfields φ andφ transform non-trivially under G, with the gauge invariant combination φφ carrying zero U(1) R charge. The singlet superfield S and the superpotential W carry unit U(1) R charges.
S provides the scalar field that drives inflation. Note that the U(1) R symmetry ensures the absence of terms proportional to S 2 , S 3 , etc in the superpotential, which otherwise could spoil the slow-roll conditions needed for implementing successful inflation. 3 From W , it is straightforward to show that the SUSY minimum corresponds to nonzero (and equal in magnitude) vacuum expectation values (VEVs) for φ andφ, while S = 0, and therefore G is broken to a subgroup H.
An inflationary scenario is realized in the early universe with φ,φ and S dis- 3 The issue of SUGRA corrections is more subtle. For W given in Eq. (1), and the minimal Kähler potential K, the flatness condition is not spoiled [2]. However, in some models such as the SU (5) case discussed here, additional fields appear (from a hidden sector as well as the visible sector), and special choices for the Kähler potential may be necessary to control SUGRA corrections [16,17]. placed from their present day minima. Thus, for S values in excess of the symmetry breaking scale M, the fields φ,φ both vanish, the gauge symmetry is restored, and a potential energy density proportional to κ 2 M 4 dominates the universe. With SUSY thus broken, there are radiative corrections from the φ-φ supermultiplets that provide logarithmic corrections to the potential which drives inflation [1]. After including the radiative corrections, the scalar potential from Eq. (1) turns out to give a scalar spectral index n s ≈ 1 − 1/N (for N ≈ 50 − 60 e-foldings) [1] and the quadrupole anisotropy δT /T [8,1] δT where The subscript l is there to emphasize the epoch of horizon crossing.
N indicates the dimensionality of the φ,φ representations, and N l ≈ 45 − 60 denotes the e-foldings needed to resolve the horizon and flatness problems. Comparison of Eq. (2) with the COBE and WMAP data [3] leads to the conclusion that the gauge symmetry breaking scale M is very close to 10 16 GeV [4], the SUSY GUT scale inferred from the evolution of the MSSM gauge couplings [18]. Thus, it is natural to try to realize the above inflationary scenario within a SUSY GUT framework.
In this paper, we will attempt to provide a realistic scenario with G = SU (5) and The minimal SU(5) model possesses a global U(1) X symmetry, where X is given by the relation B − L = X + 4 5 Y , which holds for the MSSM fields, with Y denoting the hypercharge. Note that X coincides with B − L for the SU(5) singlet right handed neutrinos. By imposing a U(1) X symmetry one prevents the appearance of superheavy ( > ∼ M GU T ) masses for the right handed neutrinos 1 i (i = 1, 2, 3). The atmospheric neutrino data and leptogenesis scenario seem to require right handed neutrino of intermediate masses ( < ∼ 10 14 − 10 15 GeV). The masses for 1 i can arise from the spontaneous breaking of U(1) X , for instance, via the non-renormalizable couplings with an extra superfield X: where the U(1) X charge of X is listed in Table I, and the dimensionless coupling y ij are of order unity or smaller. A mass of order 10 14 −10 15 GeV for the heaviest right handed neutrino can nicely explain via the seesaw mechanism the atmospheric neutrino mass scale ( ∆m 2 ATM ∼ 5 × 10 −2 eV). This can be achieved with X ∼ M GU T . Hence, it would be desirable to construct a model such that the U(1) X and SU (5) breakings are intimately linked.
If X is identified with the inflaton, as will be the case in our model, the coupling Eq. (3) can also lead to a successful leptogenesis [14]. The inflaton X exclusively decays into right handed neutrinos, and the reheat temperature T r turns out to be where M ν c is the heaviest right handed neutrino mass allowed by the kinematics of the decay process. From the gravitino constraint T r < ∼ 10 9 GeV [19], M ν c ∼ 10 10 GeV is required. Thus, the mass of X is constrained as follows: and the mass(es) of the right handed neutrino(s) lighter than X should be < ∼ 10 10 GeV. In practice, m X is taken to be of order 10 12 − 10 14 GeV, so that decay into the heaviest right handed neutrino is not allowed.
The U(1) R symmetry can forbid the bare mass term M5 H 5 H . As a consequence, within this sector, the global symmetry is enhanced to U(1) X × U(1) P Q , with U(1) X anomaly free. Motivated by this, we will impose U(1) P Q on the complete model. The U(1) R , U(1) X and U(1) P Q charge assignments for the matter, 5-plet Higgs, and the extra superfield are shown in Table I. Table I Note that with the charge assignments shown in Table I, the operators leading to baryon and lepton number violations at low energies such as 5 i 5 H , 10 i 5 j 5 k , 10 i 10 j 10 k 5 l , 10 i 10 j 10 k 5 H , 5 i 5 j 5 H 5 H , 5 i 5 H 5 H 5 H , etc. are forbidden in the absence of U(1) R , U(1) X , and U(1) P Q breakings.
To reiterate, the SU(5) model we are after should have the following features.
• U(1) R and U(1) X are suitably utilized for the desired inflation and neutrino masses.
• Inflation is associated with the spontaneous breakings of SU (5) and U(1) X at M GU T .
• Monopoles and cosmic strings do not pose cosmological problems.
• The low energy spectrum should coincide with the MSSM field content.
Consider the following trial superpotentials where Φ's and S's denote the 24-plets and singlet superfields, respectively. We normalize the SU (5) generators T i such that Tr(T i T j ) = δ ij . κ, λ are dimensionless couplings, and the superscripts ± denote U(1) X charges of ±1 for the corresponding Higgs fields. We assign zero (unit) U(1) R charges to Φ ± , Φ, S ± (S). From Eq. (1), both W 1 and W 2 appear to be viable candidates for inflation with both SU (5) and U(1) X broken at the GUT scale. However after inflation ends, in either case, unwanted cosmological defects (monopoles and/or strings) would be produced. Furthermore, the octet and triplet states (and also a pair of (3, 2) −5/6 , (3, 2) 5/6 states) contained in Φ (Φ ± ) remain 'massless' after SU (5) breaking. These states can acquire masses of order TeV when the S VEV develops after SUSY breaking. This would seriously affect the running of the MSSM gauge couplings and therefore must be avoided.
Before proceeding to the model, let us consider the Higgs superpotential W 3 [20], which we will encounter as a part of the full construction: where Φ (Φ ′ ) is a 24-plet superfield with unit (zero) U(1) R charge. Compared to minimal SU (5), an additional 24-plet Higgs Φ is introduced to realize the U (1)  For a realization of inflation and U(1) X in SU (5), we introduce some SU (5) singlets and adjoint Higgs superfields with U(1) R , U(1) X and U(1) P Q charges shown in Table II.

Table II
We identify S − with X in Table I. To make U(1) X anomaly free, one could introduce, for instance, an additional singlet (T ) and an adjoint Higgs (Φ + 1/2 ) with U(1) X charges −3 and +1, respectively. We assign a U(1) R charge of 1/2 to Φ + 1/2 , and a proper nonzero U(1) P Q charge to T . Their presence does not affect the inflationary scenario in any significant manner. The relevant renormalizable superpotential is given by  (8), one derives the F-term scalar potential: where d ijk ≡ Tr(T i T j T k ), and Φ In the presence of soft SUSY breaking terms these fields can acquire VEVs of as much as a TeV. On the other hand, S ± , P , Q, Φ + can develop VEVs such that SU(5) × U(1) X is spontaneously broken to the MSSM gauge group, where Φ + 8 is the MSSM singlet contained in Φ + ( Φ + i = 0 for i = 8), and d 888 is given by 1 √ 30 . By performing an appropriate U(1) X transformation, we can rotate away the phase of S + . We identify | Φ + 8 | with M GU T ≈ 2 − 3 × 10 16 GeV. We assume S + ∼ S − ∼ M ∼ M GU T , and P ∼ Q ∼ 10 17 GeV with U(1) R broken to Z 2 ("R-parity"). The VEVs of P and Q could be stabilized by including the soft terms and the SUGRA corrections to the scalar potential. We will see later that κ 2,3 should be of order 10 −4 − 10 −3 , while λ, α, β are of order unity.
With U(1) X broken at the M GU T scale, we get an upper bound on the right handed neutrino Majorana mass by identifying S − with X in Eq. (3): Assuming the heaviest right handed neutrino of this mass, and with a Dirac neutrino mass for the third family of order the electroweak scale, we find a light neutrino mass M 2 W /(10 14 GeV) ∼ ∆m 2 ATM ∼ 5 × 10 −2 eV. Next let us discuss how to implement doublet-triplet splitting. With some additional superfields, whose quantum numbers appear in Table III, Table III the relevant superpotential is whose presence does not affect the conclusions based on Eq. (8).
[The y's and a, b denote dimensionless couplings.] From the y c , y d terms and the soft terms, C, A, B can develop VEVs of order M, while ∆ is of order the gravitino mass scale m 3/2 . Thus, U(1) P Q is also broken at M GU T , but presumably this is acceptable within an inflationary cosmology [21]. The y e term in Eq. (14) ensures that the additional 5-plets acquire superheavy masses. The VEVs S + , Φ + 8 would make 5 H , 5 H superheavy. Thus, two fine-tunings, a = b = 10/3 S − / Φ − 8 are necessary to obtain a pair of light Higgs doublets from 5 H , 5 H . The non-renormalization theorem in SUSY ensures that such fine-tunings are stable against radiative corrections. We assume that y's and a (= b) are of order unity.
Due to the presence of SUSY breaking "A-terms," the scalar components of N, N acquire intermediate scale VEVs, ∼ m 3/2 M P ∼ 10 10 GeV. Consequently, a µ parameter (≡ y µ NN /M P ) of order m 3/2 is naturally generated [22]. The presence of U(1) P Q resolves the strong CP problem [23].
The non-zero VEVs of N, N also break the Z 2 symmetry (⊂ U(1) R ), which can lead to a cosmological domain wall problem. We therefore assume that the VEVs N , N develop before or during inflation, so that the domain walls are inflated away. With 10 i 5 j 5 H , 5 i 5 H NN S + /M 2 P , and the induced µ term in Eq. (14), the trilinear operator 10 i 5 j 5 k leading to "R-parity" violation in the MSSM is generated with a suppression factor µ 2 /(M P M GU T ). If one requires an absolutely stable LSP, one could introduce a Z 2 "matter parity." Before discussing some aspects of inflation, let us point out an important consequence of the model related to proton decay. Because of the absence of a direct coupling 5 H 5 H in the U(1) R symmetric limit, the higgsino mediated dimension five operator relevant for proton decay, 10 i 10 j 10 k 5 l , is suppressed by µ/M 2 GU T (∼ M W /M 2 GU T ), which makes it harmless. The higher dimensional operator 10 i 10 j 10 k 5 l × S − S − A /M 4 P ∼ 10 i 10 j 10 k 5 l × 10 −7 /M P is also suppressed. Thus, proton decay is expected to proceed via the superheavy gauge bosons, with life time ∼ 10 34 − 10 36 yrs.
In this model, inflation is implemented by assuming that in the early universe S > ∼ M with P ∼ 10 17 GeV. As a result, the VEVs of S + , Q, and Φ + vanish, and a vacuum energy density proportional to κ 2 1 M 4 dominates the universe, as in inflation based on Eq. (1). Inflation is driven by the radiatively generated logarithmic inflaton potential. With Φ as well as S − , Z non-zero during inflation [17,24], (1), and the SU(5) monopoles and U(1) X strings are inflated away. For simplicity, we assume that κ 2 P ∼ 1 10 κ 1 M >> Φ , S − , Z so that Eq. (2) with N = 2 still approximately holds. It turns out that with inclusion of soft SUSY breaking terms, S + , Φ + acquire VEVs during inflation that are proportional to m 3/2 , the SUSY breaking scale. As inflation ends, S + and Φ + develop GUT scale VEVs, while Φ is driven to zero, so that SU (5)  ] min , couples to the doublets in 5 H , 5 H . We note that the VEV of Ψ vanishes during and after inflation. Indeed, as S (and Z) rolls down to the origin, S + and Φ + 8 also roll down, from the origin to their present values. If Ψ remains zero throughout inflation, 4 the field S − exclusively decays into right handed neutrinos and sneutrinos via S − S − 1 i 1 j /M P and the superpotential couplings in Eq. (8). With κ 1 < ∼ 10 −2 , κ 3 < ∼ 10 −3 , S + ∼ 10 16 GeV, and Q ∼ 10 17 GeV, the inflaton S − fulfills Eq. (4).
The subsequent out of equilibrium decay of the right-handed neutrinos yields the observed baryon asymmetry via leptogenesis [14].
Before concluding, one may inquire about implementing inflation in five dimensional (5D) SU(5) models which have attracted much recent attention because of the relative ease with which the doublet-triplet problem is resolved and higgsino medi-ated dimension five proton decay is eliminated [25]. Following Refs. [26,24], inflation with δT /T ∝ (M X /M Planck ) 2 can be realized in this case, where M X (∼ 10 16 GeV) denotes the U(1) X breaking scale. Thus, the desired atmospheric neutrino mass can be realized also in 5D SU (5).
In conclusion, we have shown how inflation can be realized in SUSY SU (5) with U(1) R symmetry playing an essential role. We have also discussed how neutrino masses can be understood in this setting by exploiting a global U(1) X symmetry also broken at a scale close to M GU T . This inflationary model also possesses some interesting phenomenology. In particular, the troublesome dimension five proton decay mediated by higgsino exchange in minimal SU (5) is strongly suppressed. It would be of some interest to extend the discussion to larger GUTs such as SO (10) and E 6 .