Is Fterm hybrid inflation natural within minimal supersymmetric SO(10)?
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Abstract
We examine whether Fterm supersymmetric hybrid inflation can in a natural way be embedded within the minimal SO(10) model. We show that none of the singlets of the Standard Model symmetries in the minimal set of SO(10) representations can satisfy the conditions which are necessary for a scalar field to play the rôle of the inflaton. As a consequence, one has to introduce an extra scalar field, which, however, may spoil the naturalness of inflation within the context of SO(10). Nevertheless, if we add an extra scalar field, we are then able to construct a model that can accommodate flat directions, while it preserves the stability of the inflationary valley.
Keywords
Mass Matrix Cosmic Microwave Background Scalar Potential Cosmic String Grand Unify Theory1 Introduction
Cosmological inflation is clearly the most studied and popular scenario that can provide an answer to some of the shortcomings that plague the hot big bang model, while it predicts a spectrum of adiabatic fluctuations that can fit the cosmic microwave background (CMB) temperature anisotropies measurements [1]. Despite its success, one must, however, keep in mind that the inflationary scenario faces some problems, like the onset of inflation [2, 3, 4] and the fine tuning of the parameters in the inflationary potential so that the inflationary predictions satisfy the data [5, 6, 7, 8]. Moreover, despite the more than three decades of work on the subject, inflation still remains a paradigm in search of a model [9]. If one accepts the validity of Grand Unified Theories (GUTs) and the standard thermal history of the universe, then one finds that the universe started at a symmetric phase with high temperature and then as the universe expanded and the temperature dropped, it underwent a series of Spontaneous Broken Symmetries (SSBs), followed by phase transitions (PTs), which could have left behind topological defects, as remnants of a previous more symmetric phase. Combining GUTs with Supersymmetry (SUSY), one can consider hybrid inflationary models, which may be of the type with the Fterm (often plagued by the \(\eta \)problem, where contributions to the slowroll parameters of the order of 1, due to Planck mass suppressed corrections to the inflaton potential, may impede a sufficiently long slowroll period) or with the Dterm type (leading always to cosmic string formation, due to the breaking of an extra U(1) symmetry)^{1}. Given the plethora of precise data, arriving either from astrophysical (in particular, the CMB), or from particle physics (in particular, the large Hadron Collider (LHC)) experiments, one can examine the validity of the various inflationary models and constrain their free parameters. Moreover, one can also study whether such models can arise naturally within the framework in which they have been proposed. Following the latter approach we will study whether, within minimal supersymmetric SO(10), there is a singlet field that could play the rôle of the inflaton and thus realise an Fterm hybrid supersymmetric inflationary model [9].
In the first part of our study, we show that none of the (existing) scalar fields can satisfy the conditions necessary in order to play the rôle of the inflaton. We thus introduce, in the second part of our analysis, an extra SO(10)singlet superfield and write down the most general Higgs superpotential. We may thus propose a model of Fterm inflation embedded in SO(10) that can be in agreement with all current particle physics constraints. Certainly Fterm inflation can be realised within SO(10), but the necessity to introduce an extra singlet renders SO(10) less appealing as a gauge group describing the early evolution of our universe. We study the realisation of inflation within SO(10), because it is a wellstudied gauge group in the context of hybrid inflation.
2 Spontaneous symmetry breaking schemes within SO(10)
 Fterm hybrid inflation with superpotential [10],where \(S\) is a GUT singlet, \(\bar{\Psi }\) and \(\Psi \) are GUT Higgs fields in complex conjugate representations which lower the rank of the group by one unit when acquiring nonzero vacuum expectation values (VEVs), and \(\kappa \) and \(M\) are two constants (\(M\) has dimensions of mass) which can both be taken positive with field redefinitions. The superpotential in Eq. (1) is the most general superpotential consistent with an Rsymmetry under which \(W^\mathrm{F}\rightarrow \hbox {e}^{i\beta }W^\mathrm{F}, \bar{\Psi }\rightarrow \hbox {e}^{i\beta }\bar{\Psi }, \Psi \rightarrow \hbox {e}^{i\beta }\Psi \) and \(S\rightarrow \hbox {e}^{i\beta } S\). The scalar potential has a valley of local minima for \(S>S_\mathrm{crit}=M, \bar{\Psi }=\Psi \), and one global supersymmetric minimum at \(S=0, \bar{\Psi }=\Psi =M\). Imposing initial conditions such that \(S\gg S_\mathrm{crit}\), the fields quickly settle down the valley of local minima; the potential becomes \(V=\kappa ^2 M^4\ne 0\), supersymmetry is broken and inflation can take place. Oneloop corrections to the effective scalar potential introduce a tilt and assist the scalar field \(S\) to slowly roll down the valley of minima. When \(S\) reaches a value below \(S_\mathrm{crit}\), inflation stops by a waterfall regime and the fields settle down to the global minimum of the potential and supersymmetry gets restored.$$\begin{aligned} W^\mathrm{F}=\kappa S (\Psi \bar{\Psi }M^2), \end{aligned}$$(1)
 A series of SSBs from SO(10) down to the Standard Model (SM) times Z\(_2\) that does not generate harmful topological defects, like monopoles and domain walls, at the end of inflation. The discrete symmetry Z\(_2\) must remain unbroken down to low energies, to ensure proton stability. Following the detailed study presented in Ref. [11], the SSB cascade should take one of the following paths:$$\begin{aligned}&\mathrm{SO}(10)\rightarrow \dots \rightarrow \mathrm{G}_{3,2,2,\mathrm{B}\hbox {}\mathrm{L}}\rightarrow \mathrm{G}_\mathrm{SM}\times Z_2\nonumber \\&\quad \rightarrow \mathrm{SU}(3)_\mathrm{C} \times \mathrm{U}(1)_\mathrm{Q} \times Z_2, \end{aligned}$$(2)where the (compact) notation \(\mathrm{G}_{3,2,2,\mathrm{B}\hbox {}\mathrm{L}}\) stands for the \(\mathrm{SU(3)}_\mathrm{C}\times \mathrm{SU(2)}_\mathrm{L}\times \mathrm{SU(2)}_\mathrm{R}\times \mathrm{U(1)}_\mathrm{B}\hbox {}\mathrm{L}\), similarly \(\mathrm{G}_{3,2,1,\mathrm{B}\hbox {}\mathrm{L}}\) stands for \(\mathrm{SU(3)}_\mathrm{C}\times \mathrm{SU(2)}_\mathrm{L}\times \mathrm{U(1)}_\mathrm{R} \times \mathrm{U(1)}_\mathrm{B}\hbox {}\mathrm{L}\), and \(Z_2\) is the Rparity.$$\begin{aligned}&\mathrm{SO}(10)\rightarrow \dots \rightarrow \mathrm{G}_{3,2,1,\mathrm{B}\hbox {}\mathrm{L}}\rightarrow \mathrm{G}_\mathrm{SM}\times Z_2\nonumber \\&\quad \rightarrow \mathrm{SU}(3)_\mathrm{C} \times \mathrm{U}(1)_\mathrm{Q} \times Z_2, \end{aligned}$$(3)

Conservation of Rparity at low energies to accommodate proton lifetime stability. This requires the use of only ‘safe’ Higgs representations [12]; thus one can use \(\mathbf {10, 45, 54, 120, 126, 210}\) but not \(\mathbf {16, 144, 560}\).

Only renormalisable contributions to the superpotential.

Type I or II seesaw mechanism. This requires a \(\mathbf {\overline{126}}_\mathrm{H}\) to participate to the Yukawa couplings to fermions and appropriate Higgs couplings [13]. The type II may be more natural in the context of SO(10). The above assumptions are compatible with the framework of Ref. [11], where the formation of cosmic strings were found to be generic for a large number of SUSY GUTs. To accommodate the CMB measurements one will then have to either fine tune the parameters [5, 6, 7, 8], or to complicate the models and render the strings unstable [14]. Note that the GUT singlet \(S\) in Eq. (1) needs not be a singlet of SO(10): in fact, inflation can be triggered at any stage in the SSB cascade that finally leads to the SM. In the spirit of a minimal GUT SO(10), we will not add any SO(10) singlet, but we will rather look for the possibility that Fterm hybrid inflation is triggered during the symmetry breaking cascade initiated by a minimal GUT Higgs field content.
3 Inflation purely within minimal SO(10)
Let us consider the following two wellstudied classes of SO(10) models: the first one is based on the Higgs content \(\mathbf {210, 126, \overline{126}, 10}\) [15]; the second one focuses on realising a doublettriplet splitting and its Higgs content is \(\mathbf {54, 45, 45', 16, \overline{16}, 10, 10'}\) [16], sometimes extended by the introduction of singlets [17].
In the vein of the first class, it has been noticed that its minimal Higgs content is not fully able to account for the observed masses and mixings of the fermions when the neutrino seesaw mechanism is implemented [18]. To cure this problem, it has been proposed [19] to enlarge the model with a \(\mathbf {120}\). In what follows, we will adopt this context to perform our study, following the principle of minimal number of Higgs fields. An example of inflation embedded in the second class can be found in [20].
3.1 Higgs content and scalar superpotential

\(\Phi \) in the representation \(\mathbf {210}\). In tensor notation, it is written as a fourth rank symmetric tensor \(\Phi _{ijkl}\).

\(\Sigma \) and \(\bar{\Sigma }\) in the representations \(\mathbf {126}\) and \(\mathbf {\overline{126}}\), respectively. In tensor notation, they are written as an antisymmetric fifth rank tensor \(\Sigma _{ijklm}\).

\(H\) in the representation \(\mathbf {10}\). In tensor notation, it is written as a 10dimensional vector \(H_i\).

\(\Omega \) in the representation \(\mathbf {120}\). In tensor notation, it is written as a third rank antisymmetric tensor \(\Omega _{ijk}\).
It is clear that the only two fields that belong to conjugate representations are \(\sigma \) and \(\bar{\sigma }\); therefore, they must be the GUT Higgses that couple to the inflaton, like the superfields \(\Psi \) and \(\bar{\Psi }\) in Eq. (1). Let us address the question of whether we have an inflaton candidate. Clearly, \(a\), \(b\) and \(p\), though they all possess a coupling to \(\sigma \bar{\sigma }\), also all have a mass term, which implies that none of them can play the rôle of the inflaton field. We therefore want to find a combination of the fields \(a\), \(b\) and \(p\) that couples with \(\sigma \bar{\sigma }\) and is massless. Other conditions apply, but firstly we should ensure that there is a massless combination.
3.1.1 Case \(a_0\ne \frac{m}{2 \lambda }\)
Subcase \(b_0=0\) (and \(a_0\ne \frac{m}{2 \lambda }\))
3.1.2 Case \(a_0 = \frac{m}{2 \lambda }\)
3.2 Conditions for inflation
We now study in detail the further conditions that ensure the existence of an inflationary potential, in order to pin down the successful VEV configurations.
3.2.1 Case \(a_0 = \frac{m}{2 \lambda }\), \(b_0 = 0\), \(p_0 \ne 0\)
3.2.2 Case \(a_0 = \frac{m}{2 \lambda }\), \(b_0 = p_0 = 0\)
3.2.3 Case \(a_0 \ne \frac{m}{2 \lambda }\)
Subcase \(a_0 \ne \frac{m}{2 \lambda }\), \(b_0 = 0\)
Subcase \(a_0 \ne \frac{m}{2 \lambda }\), \(b_0 \ne 0\)
Fixing the vacuum \(b_0\) to the second solution in Eq. (20), the superpotential for \(X\) contains both a linear term in \(X\) and a coupling \(\sigma \bar{\sigma } X\), as required, but also a dangerous quadratic term. The quadratic term only vanishes when \(a_0\) is real, i.e. \(a_0 = a_0^*\), so that we will impose this condition from now on.
The vanishing of the quadratic term is, however, still not enough to ensure that the wouldbe inflaton is massless: in fact, we assumed that the inflaton \(X\) is a superposition of the chiral superfields. The condition we imposed at the beginning, makes sure that a mass in the form \(\phi _X^*\phi _X\) is zero; however, it does not ensure the vanishing of mass terms in the form \((\phi _X^*)^2 + \phi _X^2\). We numerically checked that there is no massless state once the full mass matrix, written in terms of real scalar fields, is considered in this vacuum structure. We can therefore conclude that this last subcase is excluded.
Below we will therefore assume that the minimal SO(10) is extended with the introduction of a singlet \(S\) that will play the rôle of the inflaton field.
4 Extending the minimal SO(10)
Let us then introduce an extra scalar field \(S\), which could play the rôle of the inflaton and examine whether we can find flat directions with a stable inflationary valley. We will focus on the simple case where \(S\) is a singlet of SO(10), while nonsinglets may also be used to play the rôle of the inflaton [22].
4.1 Higgs content and scalar superpotential
Of the above superpotential, we can safely neglect terms involving \(\Omega \), because it does not contain a singlet component under the SM gauge symmetries, and \(H\), since this superfield realises the electroweak SSB and has therefore a very small VEV. The third line in Eq. (22) contains the superpotential terms required for Fterm inflation. The terms in the fourth line, containing the singlet field, are potentially dangerous as they can spoil hybrid inflation by generating mass or quartic terms for the inflaton field. In the following, therefore, we will set all the extra terms containing \(S\) to zero^{3}: this shows that some tuning is necessary in order to obtain inflation.
4.2 Vacuum expectation values and superfields
4.3 Minimisation of the superpotential
4.3.1 Global minima

\(p_0=0, \ \ a_0=0, \ \ b_0=0, \ \ s_0=\frac{m_\Sigma }{\kappa }\) ;

\(p_0=0, \ \ a_0=\frac{m}{\lambda }, \ \ b_0=0\) \(s_0=\frac{m_\Sigma }{\kappa }+ 3 \frac{\eta m}{\kappa \lambda }\);

\(p_0=\frac{m}{3\lambda }\ , \ a_0=\frac{m}{3\lambda }\ , \ b_0=\pm \frac{m}{3\lambda }\ , \ s_0=\frac{m_\Sigma }{\kappa }+ \frac{4 \pm 6}{3} \frac{\eta m}{\kappa \lambda }\) ;

\(p_0=\frac{3m}{\lambda }\ , \ a_0=\frac{2m}{\lambda }\ , \ b_0=\pm \frac{im}{\lambda }\ , \ s_0=\frac{m_\Sigma }{\kappa }+ 3(1 \pm 2 i) \frac{\eta m}{\kappa \lambda }\).
4.3.2 Local minima at the onset of inflation
As a next step, one has to look for the local minimum of the potential assuming an initially large value of the VEV of the inflaton \(s\). Indeed, this is the state of the field usually assumed in chaotic inflation. To preserve the global picture of Fterm inflation, we will assume that the intermediate stage of symmetry is obtained while being in the false vacuum corresponding to \(\sigma _0 =\bar{\sigma }_0 =0\), in order to minimise the contribution from \(F_\sigma \) and \(F_{\bar{\sigma }}\) to the potential.

\(p_0=0\), \(a_0=0\), \(b_0=0\). This minimum is obviously invariant under SO(10).

\(p_0=0\), \(a_0=\frac{m}{\lambda }\), \(b_0=0\). Since \(a_0\equiv \langle \Phi (15,1,1)\rangle \), it is clear that SU(2)\(_\mathrm{L}\times \) SU(2)\(_\mathrm{R}\) is preserved by this minimum. The component of the 15 of SU(4)\(_\mathrm{C} \supset \mathrm{SU(3)}_\mathrm{C}\times U(1)_\mathrm{BL}\) that can take a VEV is the one that preserves SU(3)\(_\mathrm{C}\), which is also uncharged under \(U(1)_\mathrm{BL}\) [23]. This minimum is thus invariant under G\(_{3,2,2,\mathrm{B}\hbox {}\mathrm{L}}\).

\(p_0=\frac{3m}{\lambda }\), \(a_0=\frac{2m}{\lambda }\), \(b_0=\pm \frac{im}{\lambda }\). For the symmetries left unbroken by these minima, this case is similar to the above one (since the VEV \(p_0\) has no effect on symmetries), except that the VEV \(b_0\equiv \langle \Phi (15,1,3)\rangle \) induces the additional breaking SU(2)\(_\mathrm{R} \rightarrow \mathrm{U(1)}_\mathrm{R}\). This minimum is thus invariant under G\(_{3,2,1,\mathrm{B}\hbox {}\mathrm{L}}\).

\(p_0=\frac{m}{3\lambda }\), \(a_0=\frac{m}{3\lambda }\), \(b_0=\pm \frac{m}{3\lambda }\). A careful analysis of these minima shows that they are invariant under SU(5)\(\times \mathrm{U(1)}\) [15].
4.3.3 Stability of the inflationary valley
5 Conclusions
The inflationary paradigm has been extensively studied in the context of Supersymmetric Grand Unified Theories. Given that SO(10) is a wellstudied gauge group, we have investigated whether it can accommodate an inflationary era without the introduction of an extra scalar field to play the rôle of the inflaton. In particular, we have studied whether Fterm hybrid inflation can be incorporated in a rather natural way. We have shown that none of the scalar fields of SO(10) can play the rôle of the inflaton and one has to introduce an extra scalar field. This result may be considered as an element that spoils the naturalness of inflation within SO(10).
Adding an extra scalar field, singlet under SO(10), which could play the rôle of the inflaton, we have shown the existence of an appropriate superpotential that can have flat directions preserving the stability of the inflationary valley.
Footnotes
 1.
Fterm inflation can be studied in the context of global supersymmetry, whereas Dterm inflation must be addressed within supergravity [5].
 2.
Using the notation with indices, it is necessary to understand why other contributions to the superpotential cannot exist, namely they would not be scalars of SO(10).
 3.
After carefully studying the general case, we found that an inflationary valley can also be found for tuned values of the extra couplings \(m_\mathrm{S}\), \(\lambda _\mathrm{S}\) and \(\delta _3\). However, the minimum of the valley sits on a supersymmetric vacuum with vanishing scalar potential, therefore it cannot be used for hybrid inflation. One such solution is \(\delta _3 = \lambda ^2 \kappa M^2/(3 m^2)\), \(\lambda _\mathrm{S} =  \delta _3^3/\lambda ^2\) and \(m_\mathrm{S} = 3 m \delta _3^2/\lambda ^2\).
Notes
Acknowledgments
It is a pleasure to thank Jonathan Rocher for his collaboration on the early stages of this work.
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