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Supersymmetric Hybrid Inflation

The non-supersymmetric and supersymmetric versions of hybrid inflation are summarized. It is emphasized that supersymmetric hybrid inflation (with its extensions) is an extremely"natural"inflationary scenario since it does not require"tiny"parameters, its superpotential has the most general form allowed by the symmetries, and it can be protected against radiative or supergravity corrections. A concrete supersymmetric grand unified theory which leads to hybrid inflation, solves the mu problem and generates seesaw masses for the light neutrinos is constructed. The"reheating"which follows hybrid inflation is studied in this theory. It is shown that the gravitino constraint on the"reheat"temperature can be"naturally"satisfied. Also, the observed baryon asymmetry of the universe can be generated via a primordial leptogenesis consistently with the requirements from solar and atmospheric neutrino oscillations. Extensions of the standard supersymmetric hybrid inflationary scenario which are still consistent with all these requirements but can also avoid the cosmological disaster from the possible copious monopole production at the abrupt termination of standard hybrid inflation are constructed. They rely on utilizing the leading non-renormalizable correction to the standard hybrid inflationary superpotential and are necessary for higher unified gauge groups which predict the existence of monopoles.


THE NON-SUPERSYMMETRIC VERSION
The most important disadvantage of inflationary scenarios such as the 'new' [1] or 'chaotic' [2] ones is that they require 'tiny' coupling constants in order to reproduce the measurements of the cosmic background explorer (COBE) [3] on the cosmic microwave background radiation (CMBR). This difficulty was overcome by Linde [4] who proposed, in the context of nonsupersymmetric grand unified theories (GUTs), the hybrid inflationary scenario. The basic idea was to use two real scalar fields χ and σ instead of one that was normally used. χ provides the 'vacuum' energy density which drives inflation, while σ is the slowly varying field during inflation. This splitting of roles between two fields allows us to reproduce the observed temperature fluctuations of the CMBR with 'natural' (not too small) values of the relevant parameters in contrast to previous realizations of inflation.
The scalar potential utilized by Linde is where κ, λ are dimensionless positive coupling constants and M , m are mass parameters. The vacua lie at χ = ±2M , σ = 0. Putting m=0, for the moment, we observe that V possesses an exactly flat direction at χ = 0 with V (χ = 0, σ) = κ 2 M 4 . The mass 2 of the field χ along this flat direction is m 2 χ = −κ 2 M 2 + λ 2 σ 2 /2. So, for χ = 0 and |σ| > σ c = √ 2κM/λ, we obtain a flat valley of minima. Reintroducing m = 0, this valley acquires a non-zero slope and the system can inflate as it rolls down this valley. This scenario is called hybrid since the 'vacuum' energy density (= κ 2 M 4 ) is provided by χ, while the slowly rolling field (inflaton) is σ.
The ǫ and η criteria (see e.g., Ref. [5]) imply that, for the relevant values of parameters (see below), inflation continues until σ reaches σ c , where it terminates abruptly. It is followed by a 'waterfall', i.e., a sudden entrance into an oscillatory phase about a global minimum. Since the system can fall into either of the two minima with equal probability, topological defects (monopoles, walls or cosmic strings) are copiously produced [6] if they are predicted by the particular particle physics model employed. So, if the underlying GUT gauge symmetry breaking (by χ ) leads to the existence of monopoles or walls, we encounter a cosmological catastrophe.
The onset of hybrid inflation requires [7] that, at a cosmic time of order H −1 , H being the Hubble parameter during inflation, a region exists in the universe with size greater than about H −1 , where χ and σ happen to be almost uniform with negligible kinetic energies and values close to the bottom of the valley of minima. Such a region, at the Planck time t P = M −1 P (M P ≈ 1.22 × 10 19 GeV is the Planck mass), would have been much larger than the Planck length ℓ P = M −1 P and it is, thus, very difficult to imagine how it could emerge so homogeneous. Moreover, as it has been argued [8], the initial values (at t P ) of the fields in this region must be strongly restricted in order to obtain adequate inflation. Several possible solutions to this problem of initial conditions for hybrid inflation have been already proposed (see e.g., Refs. [9,10,11]).

THE SUPERSYMMETRIC VERSION
Hybrid inflation turns out [12] to be 'tailor made' for application to globally SUSY GUTs except that an intermediate scale mass for σ cannot be obtained in this context. Actually, all scalar fields acquire masses of order m 3/2 ∼ 1 TeV (the gravitino mass) from soft SUSY breaking.
Let us consider the renormalizable superpotential whereφ, φ is a conjugate pair of G S (the standard model gauge group) singlet left handed superfields belonging to non-trivial representations of the GUT gauge group G and reducing its rank by their vevs, and S is a gauge singlet left handed superfield. The parameters κ and M (∼ 10 16 GeV) can be made positive by field redefinitions. The vanishing of the F-term F S implies that φ φ = M 2 , whereas the D-terms vanish for | φ | = | φ |. So, the SUSY vacua lie at φ * = φ = ±M and S = 0 (from Fφ = F φ = 0). We see that W leads to the spontaneous breaking of G. The same superpotential W gives rise to hybrid inflation. The potential derived from W in Eq. (3) is D-flatness impliesφ * = e iθ φ. We take θ = 0, so that the SUSY vacua are contained. Note that W possesses a U (1) R R-symmetry:φφ →φφ, S → e iα S, W → e iα W . Actually, W is the most general renormalizable superpotential allowed by U (1) R and G. Performing appropriate G and R-transformations, we bringφ, φ, S on the real axis, i.e.,φ = φ ≡ χ/2, S ≡ σ/ √ 2 where χ, σ are normalized real scalar fields. V then takes the form in Eq.(1) with κ = λ and m = 0. So, Linde's potential for hybrid inflation is almost obtainable from SUSY GUTs but without the mass term of σ which is, however, crucial for driving the inflaton towards the vacua.
One way to generate the necessary slope along the inflationary trajectory is [13] to include the one-loop radiative corrections on this trajectory (φ = φ = 0, |S| > S c ≡ M ). In fact, SUSY breaking by the 'vacuum' energy density κ 2 M 4 along this valley causes a mass splitting in the supermultipletsφ, φ. We obtain a Dirac fermion with mass 2 equal to κ 2 |S| 2 and two complex scalars with mass 2 equal to κ 2 |S| 2 ± κ 2 M 2 . This leads to the existence of important one-loop radiative corrections to V on the inflationary valley which can be found from the Coleman-Weinberg formula [14]: where the sum extends over all helicity states i, F i and M 2 i are the fermion number and mass 2 of the ith state, and Λ is a renormalization mass scale. We find that ∆V (|S|) is given by where z = x 2 = |S| 2 /M 2 and N is the dimensionality of the representations to whichφ, φ belong. For z ≫ 1 (|S| ≫ S c ), the effective potential on this trajectory can be expanded as [13,15] Note that the slope along the inflationary valley which is provided by these radiative corrections is Λ-independent. From Eq.(6) and using the standard formalism (see e.g., Ref. [5]), we find that the quadrupole anisotropy of CMBR is with Here, N Q is the number of e-foldings suffered by our present horizon scale during inflation, and x Q = |S Q |/M , with S Q being the value of S when our present horizon scale crossed outside the inflationary horizon. For |S Q | ≫ S c , y Q = x Q (1 − 7/12x 2 Q + · · ·). Finally, from Eq.(6), one finds κ ≈ 8π The slow roll conditions (see e.g., Ref. [5]) for SUSY hybrid inflation are ǫ, |η| ≪ 1, where Note that η → −∞ as x → 1 + . However, for most relevant values of the parameters (κ ≪ 1), the slow roll conditions are violated only 'infinitesimally' close to the critical point at x = 1 (|S| = S c ). So, inflation continues practically until this point is reaches, where the 'waterfall' occurs. From the COBE [3] result, (δT /T ) Q ≈ 6.6 × 10 −6 , and eliminating x Q between Eqs. (8) and (11), we obtain M as a function of κ. For x Q → ∞, y Q → x Q and x Q y Q Λ(x 2 Q ) → 1 − . Thus, the maximal M is achieved in this limit and equals about 10 16 GeV (for N = 8, N Q ≈ 55). This value of M , although somewhat smaller than the SUSY GUT scale, is quite close to it. As a numerical example, take κ = 4 × 10 −3 which gives M ≈ 9.57 × 10 15 GeV, x Q ≈ 2.633, y Q ≈ 2.42. The slow roll conditions are violated at x − 1 ≈ 7.23 × 10 −5 , where η = −1 (ǫ ≈ 8.17 × 10 −8 at x = 1). The spectral index of density perturbations n = 1 − 6ǫ + 2η [16] is ≈ 0.985.
The SUSY hybrid inflationary scenario can be considered 'natural' for the following reasons: i. There is no need of 'tiny' coupling constants (κ ∼ 10 −3 − 10 −2 ).
ii. The superpotential W in Eq.(3) has the most general renormalizable form allowed by the gauge and R-symmetries. Moreover, the coexistence of the S and Sφφ terms in W implies that the combinationφφ is 'neutral' under all possible symmetries of W and, thus, all the nonrenormalizable terms of the form S(φφ) n , n ≥ 2, are also necessarily present in W [17]. The leading term of this type S(φφ) 2 , if its dimensionless coefficient is of order unity, can be comparable to Sφφ (recall that κ ∼ 10 −3 ) and, thus, play an important role in inflation (see Sec.3). All higher order terms of this type with n ≥ 3 give negligible contributions to the inflationary potential. The presence of U (1) R is crucial since it guarantees the linearity of W in S to all orders excluding terms such as S 2 which could generate an inflaton mass ≥ H, thereby ruining inflation by violating the slow roll conditions.
iii. SUSY guarantees that the radiative corrections do not invalidate [18] inflation, but rather provide [13] a slope along the inflationary trajectory, needed for driving the inflaton towards the SUSY vacua.
iv. Supergravity (SUGRA) corrections can be brought under control leaving inflation intact. The scalar potential in SUGRA is given [19] by where K is the Kähler potential, , and upper (lower) indices denote differentiation with respect to the scalar field φ i (φ j * ). K can be expanded as K = |S| 2 + |φ| 2 + |φ| 2 + α|S| 4 /m 2 P + · · ·, where the leading (quadratic) terms constitute the 'minimal' Kähler potential. The term |S| 2 , whose coefficient is necessarily normalized to unity, could generate a mass 2 ∼ κ 2 M 4 /m 2 P ∼ H 2 for S along the inflationary trajectory from the expansion of the exponential prefactor in Eq. (14). This would ruin inflation. Fortunately, with this particular form of W (including all the higher order terms) this mass 2 is exactly cancelled in V [12,20]. The linearity of W in S, which is guaranteed to all orders by the R-symmetry, is crucial for this cancellation to take place. This is an important property of this scheme. The |S| 4 term in K also generates a mass 2 for S via the factor (∂ 2 K/∂S∂S * ) −1 = 1 − 4α|S| 2 /m 2 P + · · · in Eq. (14), which is however not cancelled (see e.g., Ref. [21]). In order to avoid ruining the present inflationary scheme, one has then to assume [10, 15] that |α| < ∼ 10 −3 . All other higher order terms in K are harmless since they give suppressed contributions (|S| ≪ m P on the inflationary path). So, we see that a mild tuning of just one parameter is adequate for controlling SUGRA corrections. This is a great advantage of this model since in other cases tuning of infinitely many parameters is required. Moreover, note that with special forms of the Kähler potential one can solve this problem even without a mild tuning. An example is given in Ref. [11], where the dangerous mass 2 could be cancelled in the presence of fields without superpotential but with large vevs generated via D-terms. This property practically persists even in the extensions of the model we will consider in Sec.3.
In summary, for all these reasons, we consider SUSY hybrid inflation (with its extensions) as an extremely 'natural' inflationary scenario.

Hybrid Inflation in Concrete SUSY GUTs
We will now discuss how the SUSY hybrid inflationary scenario can be 'embedded' in concrete SUSY GUTs. As an example, let us consider a moderate extension of the minimal supersymmetric standard model (MSSM) based on the left-right symmetric gauge group (see Refs. [15,22,23]). The breaking of G LR to G S is achieved via a conjugate pair of SU (2) R doublet superfieldsl c , l c with B − L (baryon minus lepton number) equal to -1, 1, which acquire vevs along their right handed neutrino directionsν c H , ν c H corresponding to the superfieldsφ, φ in Sec.1.2. The (renormalizable) superpotential for the breaking of G LR is where κ, M can be made positive by field redefinitions. This superpotential leads to hybrid inflation exactly as W in Eq. (3). The quadrupole anisotropy of CMBR, (δT /T ) Q , and the coupling constant κ are given by Eqs. (8) and (11) with N = 2 sincel c , l c have two components each. An important shortcoming of MSSM is that there is no understanding of how the SUSY µ term, with the right magnitude of |µ| ∼ 10 2 − 10 3 GeV, arises. One way [24] to solve this µ problem is via a Peccei-Quinn (PQ) symmetry U (1) P Q [25], which also solves the strong CP problem. This solution is based on the observation [26] that the axion decay constant f a , which is the symmetry breaking scale of U (1) P Q , is (normally) 'intermediate' (∼ 10 11 − 10 12 GeV) and, thus, |µ| ∼ f 2 a /m P . The scale f a is, in turn, ∼ (m 3/2 m P ) 1/2 , where m 3/2 ∼ 1 TeV is the gravity-mediated soft SUSY breaking scale (gravitino mass). In order to implement this solution of the µ problem, we introduce a pair of gauge singlet superfieldsN , N with PQ charges 1, -1 and the non-renormalizable couplings λ 1 N 2 h 2 /m P , λ 2N 2 N 2 /m P in the superpotential. Here, h = (h (1) , h (2) ) is the electroweak Higgs superfield, which is a bidoublet under SU (2) L × SU (2) R , and λ 1,2 are taken positive by redefining the phases ofN , N . After SUSY breaking, theN 2 N 2 term leads to the scalar potential: where A is the dimensionless coefficient of the soft SUSY breaking term corresponding to the superpotential termN 2 N 2 and ǫ,θ, θ are the phases of A,N , N respectively. Minimization of V P Q then requires |N | = |N |, ǫ + 2θ + 2θ = π and V P Q takes the form For |A| > 4, the absolute minimum of the potential is at The µ term is generated via the N 2 h 2 superpotential term with |µ| = 2λ 1 | N | 2 /m P , which is of the right magnitude.
The potential V P Q also has a local minimum atN = N = 0, which is separated from the global PQ minimum by a sizable potential barrier preventing a successful transition from the trivial to the PQ vacuum. This situation persists at all cosmic temperatures after the 'reheating' which follows hybrid inflation, as has been shown [17] by considering the oneloop temperature corrections [27] to the potential. We are, thus, obliged to assume that, after the end of inflation, the system emerges in the PQ vacuum since, otherwise, it will be stuck for ever in the trivial vacuum.
The gauge group G LR implies the presence of right handed neutrino superfields ν c i (i = 1, 2, 3 is the family index), which form SU (2) R doublets L c i = (ν c i , e c i ) with the SU (2) L singlet charged antileptons e c i . In order to generate 'intermediate' scale masses for the ν c 's, we introduce the nonrenormalizable superpotential couplings γ il clc L c i L c i /m P (in a basis with diagonal and positive γ's). The ν c masses are then M i = 2γ i M 2 /m P (with l c , l c taken positive by a B − L transformation). Light neutrinos acquire hierarchical masses via the seesaw mechanism and, thus, cannot play the role of hot dark matter (HDM) in the universe. They are more appropriate for a universe with non-zero cosmological constant (Λ = 0) favored by recent observations [28]. The presence of HDM in such a universe is not necessary [29,30]. Note that the couplings which generate the ν c masses are also responsible for the inflaton decay (see Sec.2.1). From our discussion above, it is clear that the superpotential of the model must contain the following extra couplings: Here Note that U (1) B (and, thus, U (1) L ) invariance is automatically implied by U (1) R even if all possible non-renormalizable terms are included in the superpotential. This is due to the fact that the R charges of the products of any three color (anti)triplets exceed unity and cannot be compensated since there are no negative R charges available.
In order to avoid undesirable mixing of the components of the L's with the ones ofl c or h (2) via the allowed superpotential couplingsN N L clc , N N Lhl c , we impose a Z 2 symmetry ('lepton parity') under which L, L c change sign. This symmetry is equivalent to Z 2 'matter parity' under which L, L c , Q, Q c change sign since U (1) B is also present and contains 'baryon parity' under which Q, Q c change sign. The only superpotential terms which are permitted by U (1) P Q , U (1) R and 'matter parity' are the ones already included and LLl c l cN 2lc l c and LLl c l c hh modulo arbitrary multiplications by non-negative powers of the combinationl c l c . The vevs ofl c , l c ,N , N leave unbroken only the symmetries G S , U (1) B and 'matter parity'.

'REHEATING' AND THE GRAVITINO CONSTRAINT
A complete inflationary scenario should be followed by a successful 'reheating' satisfying the gravitino constraint [31] on the 'reheat' temperature, T r < ∼ 10 9 GeV, and generating the observed baryon asymmetry of the universe (BAU). After the end of inflation, the system falls towards the SUSY vacuum and performs damped oscillations about it. The inflaton (oscillating system) consists of the two complex scalar fields θ = (δν c and S, with equal mass m infl = √ 2κM . The oscillating fields θ and S decay into a pair of right handed neutrinos (ψ ν c i ) and sneutrinos (ν c i ) respectively via the superpotential couplings l clc L c L c and Sl c l c . The relevant Lagrangian terms are: and the common, as it turns out, decay width is given by provided that the mass M i = 2γ i M 2 /m P of the relevant ν c i satisfies the inequality M i < m infl /2.
To minimize the number of small coupling constants, we assume that so that the inflaton decays into the second heaviest right handed neutrino superfield ν c 2 with mass M 2 . The second inequality in Eq. (25) implies that y Q ≤ 2N Q /π ≈ 3.34 for N Q ≈ 55. This gives x Q < ∼ 3.5. As an example, choose x Q ≈ 1.05 (bigger values cannot lead to an adequate BAU) which yields y Q ≈ 0.28. From the COBE [3] result, we then obtain M ≈ 4.06 × 10 15 GeV, κ ≈ 4 × 10 −4 , m infl ≈ 2.3 × 10 12 GeV and M 3 ≈ 1.35 × 10 13 GeV.
The 'reheat' temperature T r , for the MSSM spectrum, is given by [15] T r ≈ 1 7 and must satisfy the gravitino constraint [31], T r < ∼ 10 9 GeV, for gravitymediated SUSY breaking with universal boundary conditions. To maximize the 'naturalness' of the model, we take the maximal value of M 2 (and, thus, γ 2 ) allowed by this constraint. This is M 2 ≈ 2.7×10 10 GeV (γ 2 ≈ 2×10 −3 ). Note that, with this M 2 , the first inequality in Eq.(25) is well satisfied.

BARYOGENESIS VIA LEPTOGENESIS
In hybrid inflationary models, it is [32] generally not so convenient to generate the observed BAU in the usual way, i.e., through the decay of superheavy color (anti)triplets. Some of the reasons are: i. Baryon number is practically conserved in most models of this type.
In some cases [33], this is a consequence of a discrete 'baryon parity' symmetry. In the left-right model under consideration, B is exactly conserved due to the presence of the R-symmetry.
ii. The gravitino constraint would require that the mass of the relevant color (anti)triplets does not exceed 10 10 GeV. This suggests strong deviations from the MSSM gauge coupling unification and possibly leads into problems with proton stability.
It is generally preferable to produce first a primordial lepton asymmetry [34] which is then partly converted into baryon asymmetry by the non-perturbative electroweak sphaleron effects [35]. Actually, in the leftright model under consideration as well as in many other models, this is the only way to generate the observed BAU since the inflaton decays into right handed neutrino superfields. The subsequent decay of these superfields into lepton (antilepton) L (L) and electroweak Higgs superfields can only produce a lepton asymmetry. It is important to ensure that this lepton asymmetry is not erased [36] by lepton number violating 2 → 2 scattering processes such as LL → h (1) * h (1) * or Lh (1) →Lh (1) * at all temperatures between T r and 100 GeV. This is automatically satisfied since the lepton asymmetry is protected [37] by SUSY at temperatures between T r and T ∼ 10 7 GeV and, for T < ∼ 10 7 GeV, these scattering processes are well out of equilibrium provided [37] m ντ < ∼ 10 eV, which readily holds in our case (see below). For MSSM spectrum, the observed BAU n B /s is related [37] to the primordial lepton asymmetry n L /s by n B /s = (−28/79)n L /s.
As already mentioned, the lepton asymmetry is produced through the decay of the superfield ν c 2 , which emerges as decay product of the inflaton. This superfield decays into electroweak Higgs and (anti)lepton superfields. The relevant one-loop diagrams are both of the vertex and self-energy type [38] with an exchange of ν c 3 . The resulting lepton asymmetry is [39] n L s where | h (1) | ≈ 174 GeV, m D 2,3 (m D 2 ≤ m D 3 ) are the 'Dirac' neutrino masses (in a basis where they are diagonal and positive), and c = cos θ, s = sin θ, with θ and δ being the rotation angle and phase which diagonalize the Majorana mass matrix of the right handed neutrinos. Note that Eq.(27) holds [40] provided that M 2 ≪ M 3 and the decay width of ν c 3 is ≪ (M 2 3 −M 2 2 )/M 2 , and both conditions are well satisfied in our model. Here, we concentrated on the two heaviest families (i = 2, 3) and ignored the first one. We were able to do this since the analysis [41] of the CHOOZ experiment [42] shows that the solar and atmospheric neutrino oscillations decouple.
The light neutrino mass matrix is given by the seesaw formula: where m D is the 'Dirac' neutrino mass matrix and M the Majorana mass matrix of right handed neutrinos. The determinant and the trace invariance of the light neutrino mass matrix imply [39] two constraints on the (asymptotic) parameters which take the form: where m 2 = m νµ and m 3 = m ντ are the (positive) eigenvalues of m ν . The µ − τ mixing angle θ 23 = θ µτ lies [39] in the range where ϕ is the rotation angle which diagonalizes the light neutrino mass matrix in the basis where the 'Dirac' mass matrix is diagonal and θ D is the 'Dirac' mixing angle (i.e., the 'unphysical' mixing angle with zero Majorana masses for the right handed neutrinos). We take m νµ ≈ 2.6 × 10 −3 eV which is the central value from the small angle MSW resolution of the solar neutrino problem [43]. The τ -neutrino mass is taken to be m ντ ≈ 7× 10 −2 eV which is the central value implied by SuperKamiokande [44]. We choose δ ≈ −π/4 to maximize −n L /s. Finally, we assume that θ D is negligible, so that maximal ν µ − ν τ mixing, which is favored by SuperKamiokande [44], corresponds to ϕ ≈ π/4. From the determinant and trace constraints and the diagonalization of m ν , we determine the value of m D 3 corresponding to maximal ν µ − ν τ mixing (ϕ ≈ π/4) for any given κ. We find that a solution for m D 3 exists provided that M 2 < ∼ 0.037M 3 . For the numerical example in Sec.2.1, we find m D 3 ≈ 8.3 GeV and m D 2 ≈ 0.98 GeV. The lepton asymmetry turns out to be n L /s ≈ −2.23 × 10 −10 and, thus, the baryogenesis constraint is satisfied. We see that, with not too 'unnatural' values of κ (≈ 4×10 −4 ) and the other relevant parameters (γ 2 ≈ 2 × 10 −3 , γ 3 ≈ 1), we can not only reproduce the results of COBE [3] but also have a successful 'reheating' satisfying the gravitino and baryogenesis requirements together with the constraints from solar and atmospheric neutrino oscillations.

Extensions of SUSY Hybrid Inflation
In trying to apply (SUSY) hybrid inflation to higher GUT gauge groups which predict the existence of monopoles, we encounter the following problem. Inflation is terminated abruptly as the system reaches the critical point on the inflationary trajectory and is followed by the 'waterfall' regime during which the scalar fieldsφ, φ develop their vevs starting from zero and the spontaneous breaking of the GUT gauge symmetry occurs. The fields φ, φ can end up at any point of the vacuum manifold with equal probability and, thus, monopoles are copiously produced [6] via the Kibble mechanism [45] leading to a cosmological catastrophe (see e.g., Ref. [46]).
One of the simplest GUT models predicting monopoles is the Pati-Salam (PS) model [47] based on the gauge group G P S = SU (4) c × SU (2) L × SU (2) R . These monopoles carry two units of 'Dirac' magnetic charge [48]. We will discuss possible solutions [6,17] of the magnetic monopole problem of hybrid inflation within the SUSY PS model, although our mechanisms can be readily extended to other semi-simple gauge groups such as the 'trinification' group SU (3) c × SU (3) L × SU (3) R , which emerges from string theory and predicts [49] monopoles with triple 'Dirac' charge, and possibly to simple gauge groups such as SO(10).

SHIFTED HYBRID INFLATION
One idea [17] for solving the monopole problem is to include into the standard superpotential for hybrid inflation the leading non-renormalizable term, which, as explained in Sec.1.2, cannot be excluded by any symmetries. If its dimensionless coefficient is of order unity, this term can be comparable with the trilinear coupling of the standard superpotential (whose coefficient is ∼ 10 −3 ). The coexistence of these terms reveals a completely new picture. In particular, there appears a non-trivial (classically) flat direction along which G P S is spontaneously broken with the appropriate Higgs fields (φ, φ) acquiring constant values. This 'shifted' flat direction can be used as inflationary trajectory with the necessary inclination obtained again from one-loop radiative corrections [13]. The termination of inflation is again abrupt followed by a 'waterfall' but no monopoles are formed in this transition since G P S is already spontaneously broken during inflation.
The spontaneous breaking of the gauge group G P S to G S is achieved via the vevs of a conjugate pair of Higgs superfields where M S ≈ 5 × 10 17 GeV is the string scale and β is taken positive for simplicity. D-flatness implies thatH c * = e iθ H c . We restrict ourselves to the direction with θ = 0 (H c * = H c ) containing the non-trivial inflationary path (see below). The scalar potential derived from δW then takes the form Defining the dimensionless variables w = |S|/M , y = |H c |/M , we obtaiñ where ξ = βM 2 /κM 2 S . This potential is a simple extension of the standard potential for SUSY hybrid inflation (which corresponds to ξ = 0) and appears in a wide class of models incorporating the leading non-renormalizable correction to the standard hybrid inflationary superpotential.
For constant w (or |S|),Ṽ in Eq. (35) has extrema at (36) Note that the first two extrema (at y 1 , y 2 ) are |S|-independent and, thus, correspond to classically flat directions, the trivial one at y 1 = 0 withṼ 1 = 1, and the non-trivial one at y 2 = 1/ √ 2ξ = constant withṼ 2 = (1/4ξ − 1) 2 , which we will use as our inflationary path. The trivial trajectory is a valley of minima for w > 1, while the non-trivial one for w > w 0 = (1/8ξ −1/2) 1/2 , which is its instability (critical) point. We take ξ < 1/4, so that w 0 > 0 and the non-trivial trajectory is destabilized before w reaches zero (the destabilization is in the chosen directionH c * = H c ). The extrema at y 3± , which are |S|-dependent and non-flat, do not exist for all values of w and ξ, since the expressions under the square roots in Eq. (36) are not always non-negative. These two extrema, at w = 0, become the SUSY vacua. The relevant SUSY vacuum (see below) corresponds to y 3− (w = 0) and, thus, the common vev v 0 ofH c , H c is given by We will now discuss the structure ofṼ and the inflationary history in the most interesting range of ξ, which is 1/4 > ξ > 1/6. For fixed w > 1, there exist two local minima at y 1 = 0 and y 2 = 1/ √ 2ξ, which corresponds to lower potential energy density, and a local maximum at y 3+ lying between the minima. As w becomes smaller than unity, the extremum at y 1 turns into a local maximum, while the extremum at y 3+ disappears. The system can freely fall into the non-trivial (desirable) trajectory at y 2 even if it started at y 1 = 0. As we further decrease w below (2 − √ 36ξ − 5) 1/2 /3 √ 2ξ, a pair of new extrema, a local minimum at y 3− and a local maximum at y 3+ , are created between y 1 and y 2 . As w crosses (1/8ξ − 1/2) 1/2 , the local maximum at y 3+ crosses y 2 becoming a local minimum. At the same time, the local minimum at y 2 turns into a local maximum and inflation along the 'shifted' trajectory is terminated with the system falling into the local minimum at y 3− which, at w = 0, develops into a SUSY vacuum. We see that, no matter where the system starts from, it always passes from the 'shifted' trajectory, where the relevant part of inflation takes place, before falling into the SUSY vacuum. So, G P S is already broken during inflation and no monopoles are produced at the 'waterfall'.
It should be noted that, after inflation, the system could fall into the minimum at y 3+ instead of the one at y 3− . This, however, does not happen since in the last e-folding or so the barrier between the minima at y 3− and y 2 is considerably reduced and the decay of the 'false vacuum' at y 2 to the minimum at y 3− is completed within a fraction of an e-folding before the y 3+ minimum even comes into existence. This transition is further accelerated by the inflationary density perturbations.
The model can be completed by adding the superpotential terms In analogy with the model of Sec.2, the first two terms are needed for solving the µ problem via a PQ symmetry, the third term represents the Yukuwa couplings, with F = (4, 2, 1), F c = (4, 1, 2) being the 'matter' superfields, and the fourth term is required for generating masses for the ν c 's (and, thus, for the ν's) and allowing the inflaton to decay. The last two terms, with G = (6, 1, 1), are novel and were introduced [50] We further impose a Z mp

SMOOTH HYBRID INFLATION
An alternative solution to the monopole problem of hybrid inflation has been proposed [6] some years ago. We will present it here within the SUSY PS model of Sec.3.1, although it can be applied to other semi-simple (and possibly some simple) gauge groups too. The idea is to impose an extra Z 2 symmetry under whichH c H c → −H c H c (say H c → −H c ). The whole structure of the model remains unaltered except that now only even powers of the combinationH c H c are allowed in the superpotential terms.
The inflationary superpotential in Eq.(33) becomes where we absorbed the dimensionless parameters κ, β in µ, M S . The resulting scalar potential V is then given bỹ where we used the dimensionless fieldsχ = χ/2(µM S ) 1/2 ,σ = σ/2(µM S ) 1/2 with χ, σ being normalized real scalar fields defined byν c H = ν c H = χ/2, S = σ/ √ 2 after rotatingν c H , ν c H , S to the real axis. The emerging picture is completely different. The flat direction atχ = 0 is now a local maximum with respect toχ for all values ofσ, and two 'new' symmetric valleys of minima appear [6,52] at They contain the SUSY vacua which lie atχ = ±1,σ = 0. It is important to note that these valleys are not classically flat. In fact, they possess an inclination already at the classical level, which can drive the inflaton towards the vacua. Thus, there is no need of radiative corrections in this case. The potential along these paths is given by [6,52] V = 48σ 4 72σ 4 1 + 1 36σ 4 1 + 1 36σ 4 The system follows, from the beginning, a particular inflationary trajectory and, thus, ends up at a particular point of the vacuum manifold leading to no production of disastrous magnetic monopoles.
The inflaton with mass m infl = 2 √ 2(µ/M S ) 1/2 µ ≈ 1.07 × 10 14 GeV decays again into ν c 2 's. One can show [52] that the gravitino and leptogenesis constraints can be satisfied with 'natural' values of the parameters together with the restrictions from solar and atmospheric neutrino oscillations and SU (4) c invariance. However, this model requires slightly higher T r 's (up to