Update on Minimal Supersymmetric Hybrid Inflation in Light of PLANCK
Abstract: The minimal supersymmetric
(or F-term) hybrid inflation is defined by a unique renormalizable
superpotential, fixed by a R-symmetry, and it employs a
canonical Kähler potential. The inflationary potential takes
into account both radiative and supergravity corrections, as well
as an important soft supersymmetry breaking term, with a mass
coefficient in the range . The latter term
assists in obtaining a scalar spectral index close to 0.96,
as strongly suggested by the PLANCK and WMAP-9yr measurements. The
minimal model predicts that the tensor-to-scalar is extremely
tiny, of order , while the spectral index running,
. If inflation is associated with
the breaking of a local symmetry, the corresponding
symmetry breaking scale is with
. This scenario is compatible with the bounds on
from cosmic strings, formed at the end of inflation from
symmetry breaking. We briefly discuss non-thermal leptogenesis
which is readily implemented in this class of models.
PACs numbers: 98.80.Cq, 12.60.Jv Published in Phys. Lett. B 725, 327 (2013)
[\fancyplain 0]\fancyplainUpdate on Minimal SUSY Hybrid Inflation in Light of PLANCK \lhead[\fancyplain]\fancyplain0 \cfoot
Supersymmetric (SUSY) hybrid inflation based on F-terms, also referred to as F-term hybrid inflation (FHI), is one of the simplest and well-motivated inflationary models susyhybrid ; hybrid . It is tied to a renormalizable superpotential uniquely determined by a global R-symmetry, does not require fine tuned parameters, and it is naturally associated with the breaking of a local symmetry, such as bl , where is the gauge group of the Minimal Supersymmetric Standard Model (MSSM) or, dvali , flipped flipped , etc. As shown in Ref. susyhybrid , the addition of radiative corrections (RCs) to the tree level inflationary potential predicts a scalar spectral index , and the microwave temperature anisotropy is proportional to , where denotes the scale of the gauge symmetry breaking. It turns out that usually is not far from . Here is the reduced Planck mass. A more complete treatment sstad2 , which incorporates supergravity (SUGRA) corrections senoguz with canonical (minimal) Kähler potential, as well as an important soft SUSY breaking term sstad1 , can yield lower values (). Recall that the minimal Kähler potential insures that the SUGRA corrections do not spoil the flatness of the potential that is required to implement FHI – reduction of by invoking non-minimal Kähler potentials is analyzed in Ref. gpp ; rlarge ; hinova .
Insisting on the simplest realization of FHI – and the one-step inflationary paradigm, cf. Ref. mhi – we wish to emphasize here that FHI is in good agreement, in a rather narrow but well-defined range of its parameters, with the latest WMAP wmap and PLANCK plin data pertaining to the CDM framework. To this end, SUGRA senoguz and soft SUSY breaking sstad1 ; sstad2 corrections are taken into account, in addition to the well-known susyhybrid RCs. The minimality of the model is justified by the fact that FHI is implemented within minimal supergravity (mSUGRA) and within a minimal extension of , obtained by promoting the pre-existing global symmetry of MSSM to a local one. As a consequence, three right-handed neutrinos, , are necessary to cancel the anomalies. The presence of leads to a natural explanation for the observed plcp baryon asymmetry of the universe (BAU) via non-thermal leptogenesis (nTL) lept , and the existence of tiny but non-zero neutrino masses. As we show, this set-up is compatible with the gravitino constraint gravitino ; kohri and the current data valle ; lisi on the neutrino oscillation parameters. It is worth mentioning that our scenario fits well with the bound plcs induced by the non-observation of the cosmic strings, formed during the phase transition. Note that strings may serve as a source strings of a controllable amount of non-gaussianity in the cosmic microwave background anisotropy.
Ii Minimal FHI Model
The minimal FHI is based on the superpotential
where , denote a pair of chiral superfields oppositely charged under , is a -singlet chiral superfield, and the parameters and are made positive by field redefinitions. is the most general renormalizable superpotential consistent with a continuous R-symmetry susyhybrid under which . The SUSY potential, , extracted (see e.g. Ref. lectures ; review ) from in Eq. (1) includes F and D-term contributions. Along the direction , the latter contribution vanishes whereas the former reads
The scalar components of the superfields are denoted by the same symbols as the corresponding superfields. Restricting ourselves to the D-flat direction, from in Eq. (2) we find that the SUSY vacuum lies at
As a consequence, leads to the spontaneous breaking of , to with SUSY unbroken.
The superpotential also gives rise to FHI since, for values of , there exist a flat direction
Thus, provides us with a constant potential energy density which can be used to implement FHI.
The Inflationary Potential.
The inflationary potential of minimal FHI, to a good approximation, can be written as
where, besides the dominant contribution in Eq. (4), includes the following contributions:
represents the RCs to originating from a mass splitting in the supermultiplets, caused by SUSY breaking along the inflationary valley susyhybrid :
where is the dimensionality of the representations to which and belong, is a renormalization scale, with being the canonically normalized inflaton field, and
where we employ the canonical Kähler potential working within mSUGRA.
where dvali ; sstad2 is the tadpole parameter which takes values comparable to , the gravitino, , mass. The soft SUSY breaking mass term for , with mass , is negligible rlarge for FHI. Also, is the dimensionless trilinear coupling, of order unity, associated with the first term of in Eq. (1). Imposing the condition , is minimized with respect to (w.r.t.) the phases and of and respectively. We further assume that remains constant during FHI.
The Inflationary Observables – Requirements.
Under the assumptions that (i) the curvature perturbation generated by is solely responsible for the one that is observed, and (ii) FHI is followed in turn by a decaying-particle, radiation and matter domination, the parameters of our model can be restricted by requiring that:
The number of e-foldings that the scale undergoes during FHI leads to a solution of the horizon and flatness problems of standard big bang cosmology. Employing standard methods hinova ; plin ; review , we can derive the relevant condition:
where is the reheat temperature after FHI, the prime denotes derivation w.r.t. , is the value of when crossed outside the horizon of FHI, and is the value of at the end of FHI. This coincides with either the critical point appearing in the particle spectrum of system during FHI – see Eq. (\theparentequationb) – or the value for which one of the slow-roll parameters review
exceeds unity. In our scheme, we exclusively find . Since the resulting values are sizably larger than – see next section – we do not expect the production of extra e-foldings during the waterfall regime, which in our case turns out to be nearly instantaneous – cf. Ref. bjorn .
The (scalar) spectral index , its running, , and the scalar-to-tensor ratio, , which are given by
at 95 confidence level (c.l.).
Here, we adapt to our set-up the results of the simulations for the abelian Higgs model following Ref. markjones , , with being the gauge coupling constant close to . Note that the presence of strings does not anymore mark allow closer to unity.
The investigation of our model depends on the parameters:
In our computation, we use as input parameters and , and fix , as suggested by our results in Sec. III. Variation of over orders of magnitude is not expected to significantly alter our findings – see Eq. (3). We then restrict and so that Eqs. (3) and (5) are fulfilled. Using Eqs. (\theparentequationa) and (\theparentequationb), we can extract the values for and , thereby testing our model against the observational data of Eqs. (\theparentequationa) and (\theparentequationb).
Our results are presented in Figs. 1, 2 and 3 taking (solid lines), (dashed lines), (dot-dashed lines), and (double dot-dashed lines). In Figs. 1 and 2 the observationally compatible region of Eq. (\theparentequationa) is also indicated by the horizontal (in Fig. 1) or vertical (in Fig. 2) lines. For the sake of clarity, we do not show solutions with – cf. Ref. sstad2 – which are totally excluded by Eq. (\theparentequationa).
From Fig. 1, where we depict versus (a) and (b), we note that, for and , and progressively – for and , – dominates in Eq. (5), and drives to values close to or larger than , independently of the selected values. On the other hand, for , starts becoming comparable to and succeeds in reconciling with Eq. (\theparentequationa) for well defined (and ) values that are related to the chosen . Actually, for the allowed , we find that , whereas turns out to be totally negligible. Fixing to its central value in Eq. (\theparentequationa), we display in Table 1 the values for corresponding to the values employed in Figs. 1-3.
From our numerical computations we observe that, in the regime with acceptable values, the required by Eqs. (3) and (5) becomes comparable to , and in Eq. (\theparentequationb) can be approximated as hinova
Moreover, in the vicinity of , develops a local maximum at allowing for FHI of hilltop type lofti to take place. As a consequence, , and therefore in Eq. (4) and in Eq. (\theparentequationb) – see Fig. 2-(b) –, decrease sharply (enhancing ), whereas (or ) increases adequately, thereby lowering within the range of Eq. (\theparentequationa). In particular, for constant , the lower the value for we wish to attain, the closer we must set to . To quantify the amount of these tunings, we define the quantities
and list their resulting values in Table 1. From there, we conclude that the required tuning is at a few percent level, since . Values of well below are less desirable from this point of view. For comparison, we mention that for , we get , i.e., increases with whereas the maximum disappears. From Table 1, we note that and decrease with and , too.
In Fig. 2-(a) and Fig. 2-(b) respectively we display the predictions of our model for and . Corresponding to the values within Eq. (\theparentequationa), turns out to be of order . On the contrast, is extremely tiny, of order , and therefore far outside the reach of PLANCK and other contemporary experiments. For the preferred values, we observe that and increase with whereas for constant , , and increase with . For the values used in Fig. 2 and with , our predictions are summarized in Table 1.
The dependence of on within our model is shown in Fig. 3. We remark that mostly decreases with . For low enough values, there is region where we get two values consistent with Eqs. (3) and (5). Comparing Fig. 3 with Fig. 1-(b), we can easily conclude that the latter solution is consistent with Eq. (\theparentequationa). The values displayed in this figure are fully compatible with the upper bound arising from Eq. (1). Although these values lie somewhat below , the unification of gauge coupling constants within MSSM remains intact since the gauge boson associated with the spontaneous breaking is neutral under , and so it does not contribute to the relevant renormalization group (RG) running.
In order to highlight the differences of the various possible solutions obtained at low values, we present in Fig. 4 the variation of as a function of for the same and and two different values compatible with Eqs. (3) and (5). Namely, we take , and  yielding  with  – gray [light gray] line. The corresponding and values are also shown. As we anticipated above, in the first case, develops a maximum at decreasing thereby at an acceptable level – we get as shown in Table 1. Needless to say that, in both cases, turns out to be bounded from below for large values and, therefore, no complications arise in the realization of the inflationary dynamics.
As inferred from Fig. 1, for any we can conveniently adjust , so that Eq. (\theparentequationa) is fulfilled. Working in this direction, we delineate the (lightly gray) region in the  plane allowed by all the imposed constraints – see Fig. 5-(a) [Fig. 5-(b)]. We also display by solid lines the allowed contours for . We do not consider values lower than , since they would be less natural from the point of view of both SUSY breaking and the ’s and ’s encountered – see Table 1. The boundaries of the allowed areas in Fig. 5 are determined by the dashed [dot-dashed] lines corresponding to the lower [upper] bound on in Eq. (\theparentequationa). In these regions we obtain which are compatible with Eq. (1). On the other hand, these regions are not consistent with the most stringent (although controversial sanidas ) constraint pta imposed by the limit on the stochastic gravitational wave background from the European Pulsar Timing Array. These latter results depend on assumptions regarding string loop formation and the gravitational waves emission. The bounds on from , are totally avoided if we implement FHI within dvali or flipped flipped , with or respectively in Eq. (\theparentequationa), which do not lead to the production of any cosmic defect – for a more complete discussion involving flipped and the corresponding values, see second paper in sstad2 .
The values are consistent with Eq. (1) according to which . The maximal values for and are respectively encountered in the upper left and right corners of the allowed region in Fig. 5-(b). In the lower left [right] corner of that area, we obtain the lowest possible . Also, ranges between and whereas varies between and .
Iii Non-Thermal Leptogenesis
As FHI ends, crosses , thereby destabilizing the system which leads to a stage of tachyonic preheating as described in Ref. buch . Soon afterwards, the inflaton system (IS) settles into a phase of damped oscillations about the SUSY vacuum, eventually decaying and reheating the universe. Note that the IS consists of the two complex scalar fields and , where and . To ensure the decay of the IS and implement the see-saw mechanism for the generation of the light neutrino masses, we allow for the following superpotential terms:
where  have charge of and R charge . denotes the -th generation doublet left-handed lepton superfields, and is the doublet Higgs superfield which couples to the up quark superfields.
At the SUSY vacuum, Eq. (3), and acquire their v.e.vs, thereby providing masses to the IS and ’s,
The predominant decay channels of and are to (kinematically allowed) bosonic and fermionic ’s respectively via tree-level couplings derived from Eqs. (1) and (1) – see e.g. Ref. lectures – with almost the same decay width gmb
We assume here that the problem of MSSM is resolved as suggested in Ref. flipped ; rsym , rather than by invoking the mechanism of Ref. dvali which would open new and efficient decay channels for . The SUGRA-induced Idecay decay channels are negligible in our set-up, with the and values in Eq. (\theparentequationa). The resulting reheat temperature is given by quin
where counts the MSSM effective number of relativistic degrees of freedom at temperature .
For , the out-of-equilibrium decay of generates a lepton-number asymmetry (per decay), . The resulting lepton-number asymmetry is partially converted through sphaleron effects into a yield of the observed BAU:
being the branching ratio of IS to . The quantity can be expressed in terms of the Dirac masses of , , arising from the second term of Eq. (1).
where we take into account only thermal production of , and assume that is much heavier than the MSSM gauginos – the case of CDM was recently analyzed in Ref. buch .
The success of our post-inflationary scenario can be judged, if, in addition to the constraints of Sec. II, it is consistent with the following requirements:
The bounds on :
for some ’s. The first bound comes from the needed perturbativity of ’s in Eq. (1), i.e. . The second inequality is applied to avoid any erasure of the produced due to mediated inverse decays and scatterings lsenoguz . Finally, the last bound above ensures a kinematically allowed decay of the IS for some ’s.
Constraints from Neutrino Physics. We take as inputs the best-fit values valle – see also Ref. lisi – on the neutrino mass-squared differences, and , on the mixing angles, , , and and the CP-violating Dirac phase for normal [inverted] ordered (NO [IO]) neutrino masses, ’s. The sum of ’s is bounded from above by the data wmap ; plcp , at 95% c.l.
The bounds on imposed kohri by successful BBN:
Here we consider the conservative case where decays with a tiny hadronic branching ratio.
|Low Scale Parameters|
|Decay channels of the Inflaton System, I|
|Resulting and -Yield|
The inflationary requirements of Sec. II restrict and in the very narrow range presented in Eq. (\theparentequationa). As a consequence, the mass of IS given by Eq. (2), is confined to the range , and its variation is not expected to decisively influence our results on . For this reason, throughout our analysis here we use the central value , corresponding to the second row of Table 1.
On the other hand, (and ) also depend on the masses of into which the IS decays. Following the bottom-up approach – see Sec. IVB of Ref. nMCI –, we find the ’s by using as inputs the ’s, a reference mass of the ’s – for NO ’s, or for IO ’s –, the two Majorana phases and of the MNS matrix, and the best-fit values mentioned above for the low energy parameters of neutrino physics. In our numerical code, we also estimate, following Ref. running , the RG evolved values of the latter parameters at the scale of nTL, , by considering the MSSM with as an effective theory between and the SUSY-breaking scale, . We evaluate the ’s at , and we neglect any possible running of the ’s and ’s. Therefore, we present their values at .
Our results are displayed in Table 2 taking some representative values of the parameters which yield the correct , as dictated by Eq. (8). We consider NO (cases A and B), degenerate (cases C, D and E) and IO (cases F and G) ’s. In all cases the current limit (see point 2 above) on the sum of ’s is safely met – the case D approaches it. The gauge group adopted here, , does not predict any relation between the Yukawa couplings constants entering the second term of Eq. (1) and the other Yukawa couplings in the MSSM. As a consequence, the ’s are free parameters. However, for the sake of comparison, for case A, we take , and in case B, we also set , where and denote the masses of the top and charm quark respectively. We observe that in all cases . This is done, in order to fulfill the second inequality in Eq. (7), given that heavily influences . Note that such an adjustment requires theoretical motivation, if the gauge group is or flipped – cf. Ref. lsenoguz .
From Table 2 we observe that with NO or IO ’s, the resulting ’s are also hierarchical. With degenerate ’s, the resulting ’s are closer to one another. Therefore, in the latter case more IS-decay channels are available, whereas for cases A and G only a single decay channel is open. In all other cases, the dominant contributions to arise from . In Table 2 we also display, for comparison, the -yield with () or without () taking into account the RG effects. We observe that the two results are mostly close to each other with some discrepancies appearing for degenerate and IO ’s. Shown also are values for , the majority of which are close to , and the corresponding ’s, which are consistent with Eq. (9) mostly for . These large values can be comfortably tolerated with the ’s appearing in Fig. 5 for – see the definition of below Eq. (2). From the perspective of constraint, case A turns out to be the most promising.
Inspired by the recently released WMAP and PLANCK results for the inflationary observables, we have reviewed and updated the predictions arising from a minimal model of SUSY (F-term) hybrid inflation, also referred to as FHI. In this set-up susyhybrid , FHI is based on a unique renormalizable superpotential, employs a canonical Kähler potential, and is associated with a superheavy phase transition. As shown in Ref. sstad2 , and verified by us here, to achieve values lower than , one should include in the inflationary potential the soft SUSY breaking tadpole term, with the SUSY breaking mass parameter values in the range . Fixing to its central value, the dimensionless coupling constant, the symmetry breaking scale, and the inflationary parameters and are respectively given by , , and . The cosmic strings, formed at the end of FHI, have tension ranging from to and may be accessible to future observations. We have also briefly discussed the reheat temperature, gravitino constraints and non-thermal leptogenesis taking into account updated values for the neutrino oscillation parameters.
Acknowledgments.Q.S. acknowledges support by the DOE grant No. DE-FG02-12ER41808. We would like to thank W. Buchmüller, M. Hindmarsh, A. Mazumdar and K. Schmitz for useful discussions.
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