Higher-Dimensional Origin of Heavy Sneutrino Domination and Low-Scale Leptogenesis

If the expectation value of the right-handed (rhd) sneutrino comes to dominate the universe, its decay naturally leads to successful leptogenesis, as well as significant dilution of dangerous inflationary relics, such as the gravitino. The resulting baryon asymmetry is independent of other cosmological initial conditions. This attractive variant of leptogenesis requires at least one of the rhd neutrinos to have small Yukawa coupling and to have mass ~ 10^6 GeV, much smaller than the grand unified (GUT) scale. We show that these features naturally arise in the context of independently motivated and successful 5d orbifold GUTs with inverse-GUT-scale-sized extra dimensions. Rhd neutrinos are realized as bulk fields N_i with 5d bulk masses, while Yukawa couplings and lepton-number-violating masses for the N_i are localized at the SM boundary. The exponential suppression of the would-be N_i zero-modes leads to the desired small 4d Yukawa couplings and small masses for the rhd neutrino states. The see-saw prediction for the lhd neutrino mass scale is automatically maintained. We show that this realization of rhd neutrinos is nicely accommodated within an attractive orbifold-GUT flavour model, where all flavour hierarchies have a geometrical origin.


Basics of the scenario
With growing experimental evidence for neutrino masses in a range that is consistent with a GUT-scale-based see-saw mechanism [1], leptogenesis has become the standard scenario for the generation of the baryon asymmetry of the universe. In the original proposal [2], the heavy rhd neutrinos decay in an out-of-equilibrium fashion once the universe has cooled to a temperature below their mass scale. The resulting lepton number is then converted to baryon number by standard model (SM) sphaleron processes (for a recent review see, e.g., [3]). Alternatively, in a supersymmetric theory, lepton number can be generated by the decay of a condensate of the scalar componentÑ of the rhd neutrino superfield N [4]. In particular, a recent detailed analysis [5] of this scenario has shown that, ifÑ comes to dominate the universe, its decay can naturally produce the required lepton asymmetry independently of other cosmological initial conditions. At the same time, the number density of dangerous inflationary relics, such as the gravitino, is significantly diluted.
This cosmologically attractive variant of leptogenesis requires at least one of the rhd neutrino masses to be very small compared with the GUT scale, ∼ 10 6 GeV, and the corresponding Yukawa coupling to be suppressed. In the present paper, we show that such a situation arises naturally in the context of independently motivated higher-dimensional GUTs (with inverse-GUT-scale-sized extra dimensions). Before embarking, in Sects. 2 and 3, upon a detailed discussion of the orbifold-GUT model and of the cosmology, we now explain the fundamentals of our scenario.
To be specific, we will formulate our ideas in the framework of supersymmetric SU (5) orbifold GUTs [6,7,8,9] in 5 dimensions. These theories are attractive because they incorporate the success of MSSM gauge-coupling unification [8,9], while providing natural doublet-triplet splitting as well as suppressed proton decay [6,7,8,9]. Moreover, a three generation model with a geometrical origin of hierarchical Yukawa couplings and see-saw neutrinos can easily be realized [10].
The starting point for these models is a 5d super Yang-Mills theory compactified on an interval with coordinate y ∈ [0, l]. At the y = 0 boundary (the 'SU(5) brane') the 5d gauge symmetry is unbroken by boundary conditions, while at y = l (the 'SM brane'), the boundary conditions on the gauge fields explicitly break the 5d SU(5) down to the SM gauge group. Below the compactification scale M c ≡ 1/l, one has a 4d effective field theory with SM gauge group (and N=1 supersymmetry).
A basic property of such models is that the 'bulk', y ∈ (0, l), is moderately large compared to the fundamental 5d Planck (or UV cutoff) length. Arguments pointing to this conclusion include the weakness of the effective 4d unified coupling, the 'observed' smallness of GUT-scale threshold corrections, and the flavour hierarchies among the generations (see e.g. [8,9,11]). As we discuss in Sect. 2, an alternative way of quantifying the size of the bulk follows from the requirement that gauge coupling unification (including the KK-mode corrected logarithmic running above M c ) occurs at the 5d Planck scale. Concretely this argument favours an orbifold GUT setup with Ml ≃ 300 and M ≃ 1.4 × 10 17 GeV, where M is the reduced 5d Planck mass. This is in accord with the size of the bulk deduced from other arguments.
If a bulk mass m, odd under 5d parity, is introduced for a bulk matter field, its zero mode develops an exponential profile ∼ exp[−ym] [12] (see also [13,11] and [14]). Thus, depending on the sign of m, zero modes can be strongly peaked at either brane. If one of the three SM-singlet rhd neutrino fields N i is exponentially peaked at the SU(5) brane, while lepton number violating mass terms ∼ N 2 i are allowed only at the SM brane, an exponential suppression of both the 4d rhd neutrino mass and Yukawa coupling naturally arises. The crucial observation is that the light neutrino masses are not affected even if one or more of the N i are such bulk fields with arbitrary bulk profile. This is clear because the N i zero modes receive only their kinetic term from the bulk, while their effective 4d mass and Yukawa coupling come from the brane. When the N i are integrated out, their kinetic term plays no role and thus it is irrelevant whether they are brane or bulk fields. Therefore the traditional see-saw prediction for the lhd neutrino mass scale is maintained. However, the rhd sneutrino mass scale is exponentially suppressed, as are its Yukawa couplings, and thus decay width. These are the new features that allow us to realise the attractive scenario of sneutrino (Ñ) dominated cosmology and leptogenesis.
In Sect. 2 we present a more detailed motivation and quantitative analysis of the basic orbifold picture of neutrino masses and interactions, in particular a demonstration that it can be successfully embedded in a full flavour model. Specifically, both Higgs doublets and the three 5's of SU(5) (denoted by F i ) are localized at the SM brane, while the three 10's (denoted by T i ) are bulk fields. The quark and lepton mass hierarchies are generated by the bulk profiles of the T i s.
However, we emphasise that our neutrino mass construction is quite generic and does not rely on the details of the specific SU(5) model worked out in the rest of this paper. The crucial ingredient is a 5d, or higher-dimensional theory compactified on an interval with Yukawa couplings and lepton-number-violating neutrino masses localized at one of the boundaries. The exponential suppression of zero-mode wave functions at that boundary generates both the fermion mass hierarchy and the desired light rhd neutrinos.
The cosmology of the above model of neutrinos has many attractive aspects. In particular, over a wide parameter region it leads to theÑ dominated early universe of [5], asÑ has an exponentially enhanced life-time. In more detail, if the initial value of |Ñ| is of the order of M, a natural circumstance, thenÑ will come to dominate the universe for inflationary reheating temperatures T R > ∼ 10 9 GeV. Moreover, if T R is varied between ∼ 10 9 GeV and ∼ 10 12 GeV, the gravitino number density in the late universe remains fixed at the level corresponding to T R ∼ 10 9 GeV. This attractive feature ofÑ dominated cosmology is due to the entropy produced byÑ decay. Finally, the decay of N produces the lepton-number asymmetry.
This cosmology is a fascinating possibility since most of the important physical parameters in the present universe, such as baryon-number asymmetry, entropy (and, as we later discuss, even spectrum of density fluctuations), are determined by the nature of the scalar partner of the lightest rhd neutrino. In Sect. 3, we provide a more detailed discussion of this cosmology, while some further possibilities, together with our conclusions, are contained in Sect. 4.

The flavour model
Consistency of the orbifold GUT framework requires M c = 1/l to be significantly smaller than the UV scale M of the 5d gauge theory. To be more specific, the (reduced) Planck masses in 4d (M P ) and in 5d (M), are related by and we demand gauge coupling unification at the fundamental scale M. In spite of the UV sensitivity of the non-renormalizable 5d theory, the differences of inverse SM gauge couplings 3) continue to run logarithmically above M c [8,9] because these differences are only sensitive to the SU(5)-breaking SM brane. In the context of the minimal model of [9], where the Higgs-doublets are localized at the SM brane, this 'differential running' [15] comes entirely from the gauge sector. With the effective SUSY breaking scale set to m Z , we have where  [16], the model is fixed by requiring the precision of simultaneous 1-2 and 2-3 unification to be better than with conventional MSSM running.) The above discussion provides us with a motivation for an orbifold GUT setup with M ≃ 1.4 × 10 17 GeV and with the small parameter ε 2 ≡ 1/(Ml) ≃ 1/300. Flavour is described by introducing the three F i fields and the two Higgs doublets H u and H d on the SM brane (recall that, because of the reduced symmetry of the SM brane, there is no need for Higgs triplets), while allowing the T i to propagate in the bulk. 1 This large disparity between T 's and F 's is the geometric origin of small quark and large lepton mixing (cf. [17] and [10]). If a bulk mass m, odd under 5d parity, is introduced for a 5d hypermultiplet, its zero mode develops an exponential profile ∼ exp[−ym]. Thus, depending on the sign of m, zero modes can be strongly peaked at either brane. In particular, this allows for a dynamical realization of SM-brane fields with quantum numbers appropriate for an SU(5) representation (e.g., the Higgs doublets and the F i above). We will use this additional tool to realize the fermion mass hierarchy by appropriately localizing the T i .
An understanding of the observed hierarchies in the fermion masses and mixings emerges naturally if the bulk mass of T 3 is sufficiently large and negative, m < 0 (so that, for all practical purposes, T 3 is a SM brane field), while T 2 has vanishing bulk mass (flat zero-mode) and T 1 has a finite bulk mass m > 0 (its zero-mode therefore being suppressed at the SM brane).
Concretely, the primordial and unstructured O(1) Yukawa couplings λ at the SM brane are rescaled as for each participating bulk field with bulk mass m. This rescaling follows from the 4d canonical normalization of the 5d kinetic term and the exponentially suppressed zeromode field value at the SM brane. Applying this to T 2 , one finds that this field enters Yukawa interactions with a suppression factor ε. The analogous suppression factor for T 1 depends on m and becomes ∼ ε 2 for the choice ml ≃ 3.9. This leads to the following realistic Yukawa matrix structure for the two effective 4d interactions H u T T λ T T T and with unknown O(1) factors multiplying each entry. 2 It is known that this Yukawa coupling hierarchy also gives rise to an approximately correct CKM structure. The top Yukawa coupling is naturally O(1). The required relative suppression of down-type Yukawa couplings can be realized either by going to large tanβ or by slightly decreasing the strength with which H d is peaked at the SM brane.
The construction presented so far can be summarized as follows. By identifying the 5d Planck mass with the unification scale, we have argued for a relative bulk size characterized by ε ∼ 1/ √ Ml ∼ 1/ √ 300. If all fields except T 1 and T 2 are localizing at the SM brane, this bulk suppression factor beautifully explains the mass hierarchy between the two heavier generations [11]. To explain the extreme lightness of the first generation, we had to give T 1 a bulk profile exponentially suppressed at the SM brane using the additional tool of bulk masses. With this tool in hand, rhd neutrino singlets can easily acquire the exponentially suppressed 4d masses and couplings required for thẽ N dominated universe. Now we discuss the rhd neutrinos in more detail. Consider introducing three neutrino fields N i at the SM brane. Given a Majorana mass matrix M N,ij with O(M) entries and O(1) Yukawa couplings between N i , F i and H u , the conventional see-saw mechanism leads to a light neutrino mass scale |H u | 2 /M ≃ 2 × 10 −4 eV. In the present scenario, such a small mass scale is welcome since it ensures the out-of-equilibrium decay ofÑ (see Sect. 3). The observed neutrino oscillations, which require a somewhat larger light neutrino mass scale, can be accommodated by assuming that M N has two slightly suppressed eigenvalues. (A concrete example of such a suppression mechanism will be provided shortly.) As discussed in Sect. 1, light neutrino masses are not affected if one or more of the N i are promoted to bulk fields. To be specific, let us declare N 1 to be a bulk field with bulk mass m 1 and effective 4d mass (cf. Sect. 5 of [10]) Due to the exponential suppression factor, the desired small value of M 1 is easily realized: for example, M 1 ≃ 3 × 10 6 GeV for m 1 l ≃ 11. While this concludes the description of our basic flavour model with a naturally light rhd neutrino, several open issues deserve further discussion.
Firstly, we need to enhance two of the light neutrino masses. For example, one could introduce a Froggatt-Nielsen U(1), broken by two charge-(±1) fields with vacuum expectation values |Φ ± | where |Φ ± |/M ≃ η ≪ 1 [18]. With charge assignments (0,−1,−1) and (1,1,1) for the N i and F i respectively, one obtains the following structures for the Yukawa matrix λ N in H u F T i λ N,ij N j and the mass matrix: It is easy to convince oneself that all entries of the resulting light neutrino mass matrix m ν ≃ λ N M −1 N λ T N × |H u | 2 are of the order η −2 |H u | 2 /M. This also sets the scale for two of the eigenvalues. Although the remaining eigenvalue is suppressed to η 2 |H u | 2 /M, all three mixing angles are generically large. 3 In our setup, realistic neutrino phenomenology requires η ∼ 10 −1 . Furthermore, assigning a suitable U(1) charge to H d provides an alternative way to realize suppressed down-type masses. Let us finally argue why the family-symmetry should be broken in the U(1) charge assignment of the N i . One possibility is to demand vanishing U(1) charges for all bulk fields. Alternatively, one could replace the U(1) with a Z 3 and then note that, while the cancellation of the mixed Z 3 -SU(5) anomaly forces all three F i to have the same charge, the N i charges remain unrestricted.
Secondly, it is necessary to forbid both parity-even bulk masses as well as SU(5)brane-localized mass terms for the rhd neutrinos. Following [10], this can be done by gauging U(1) χ (named as in [20]), defined by SU(5)×U(1) χ ⊂ SO(10). SinceÑ domination requires a large initial value ofÑ 1 , the D-term potential forÑ 1 has to be suppressed. This can be achieved by dynamically breaking U(1) χ at the high scale M in 5d. A surviving discrete subgroup will be sufficient to forbid the dangerous mass operators. In addition, it is natural that U(1) χ is broken by orbifolding at the SM brane [10], making it the only possible location for the required lepton-number violating mass term. Furthermore, we would like to comment on the relation of our method of generating the fermion mass hierarchy and the light N 1 field to the familiar Froggatt-Nielsen approach. Certainly the assignment of bulk masses to different sets of fields resembles the assignment of U(1) charges. This similarity becomes even more pronounced if the bulk masses are dynamically realized by expectation values of U(1) fields with Fayet-Iliopoulos terms at the boundary (see, e.g., [21]). However, especially in the case of the large suppression factor needed for N 1 , it is a significant advantage that the bulk mass effect is exponential rather than power-like. Furthermore, there are crucial qualitative differences in the resulting phenomenology. For example, higher-order Kähler-terms involving T † i T i together with the SUSY-breaking spurion, which can lead to dangerous flavour violation, are unrestricted by U(1) symmetries. In our case, if SUSY breaking is localized at the SM brane, such terms will be geometrically suppressed for the first two generations. The argument extends to the F i if some of them are promoted to bulk fields.
Finally, note that the bulk masses used in the above construction are significantly smaller than the fundamental 5d scale M. This may follow naturally if bulk masses come from expectation values of weakly coupled U(1) fields. Alternatively, one may imagine the 5d theory to descend from a 6d theory, where bulk masses are forbidden, so that 5d bulk masses are due to small, non-perturbative effects arising in the 6d to 5d compactification process.

Cosmology
Let us turn to the discussion of cosmology. It is a reasonable assumption that the scalar partner of at least one of the right-handed neutrinos has, during inflation, an expectation value of the order of the cutoff scale M. The reason for this is that higher-dimension operators in the Kähler potential link the inflationary sector, in particular the superfield whose F -term or D-term gives rise to the non-zero vacuum energy, and the rhd neutrino superfields. (Higher order superpotential terms can be forbidden by a continuous or discrete symmetry acting on the superfield N 1 .) This leads to a contribution to the (mass) 2 of the sneutrino of order m 2 eff ∼ (H inf ) 2 , where H inf is the inflationary expansion rate. The sign depends upon the unknown Kähler operator coefficient. If m 2 eff < 0, theñ N 1 gains an expectation value only limited by yet higher-order terms in the Kähler potential, suppressed by powers of M. (Note that the parametrically small inflationary F -or D-term expectation value multiplies the entire set of higher dimension operators which lead to a potential forÑ 1 .) Here we have assumed that H inf is larger than the mass of the right-handed neutrino.
As described in the previous sections, for the lightest rhd sneutrino fieldÑ 1 both its mass M 1 (cf. Eq. 5) and its effective 4d Yukawa coupling, are exponentially suppressed. When the expansion rate H after the end of inflation decreases below M 1 ,Ñ 1 starts coherently oscillating. Given a condition on the postinflationary reheating temperature T R (to be discussed below), the oscillation energy dominates the energy density of the early universe, and its decay produces the baryon asymmetry observed today without any cosmological difficulty.
Here, δ ef f is an effective CP violating phase. This lepton asymmetry is converted into a combined baryon and lepton asymmetry through non-perturbative electroweak sphaleron effects. A crucial observation of ref. [5] is that the final baryon asymmetry is determined by the reheating temperature, T N 1 , of theÑ 1 decay once it dominates the energy density of the early universe. The net baryon to entropy ratio is given by [5] n B /s ≃ (8 where n B and s are baryon-number and entropy densities, respectively. The observed baryon asymmetry n B /s ≃ (0.4 − 1) × 10 −10 is obtained by taking T N 1 ≃ 2 × 10 6 GeV for δ ef f ≃ 1. On the other hand, the reheating temperature T N 1 due toÑ 1 decay is given by We see that the desired reheating temperature T N 1 ≃ 2 × 10 6 GeV is obtained for m 1 l = 11, where we have used Eqs. (5) and (7). Notice that the rhd neutrino mass M 1 ≃ 3 × 10 6 GeV > T N 1 , and hence the out-of-equilibrium condition forÑ 1 decay is automatically satisfied.
Let us now discuss the condition for theÑ 1 domination in the early universe. Since the initial value of |Ñ 1 | ≃ 10 17 GeV, the energy density of the coherentÑ 1 oscillation, at the start of this oscillation, is a minor component of the total density. However, if theÑ 1 lifetime is sufficiently longer than that of the inflaton, it can dominate the early universe since the energy density of the radiation resulting from the inflaton decay dilutes faster than the energy density of the coherent oscillation. Thus, the condition for theÑ 1 domination is translated to an upper limit on the inflaton lifetime for a givenÑ 1 lifetime. Written in terms of the post-inflationary reheating temperature, T R , this condition is which is easily satisfied in a variety of inflationary models.
The post-inflationary reheating temperature must also satisfy an upper bound so as to avoid the over-production of gravitinos. In the standard cosmology (withoutÑ 1 domination) the upper bound is determined to be T R < 10 10 GeV for a gravitino mass ∼ 1 TeV [23] (for a recent analysis see also [24]), giving a stringent restriction on inflationary models.
However, withÑ-dominated cosmology, only a weaker constraint applies. The reason for this is thatÑ 1 decay reheats the universe once more, and the associated entropy production dilutes substantially the density of earlier-produced gravitinos. In detail, start from a situation where T R is at its lower bound given by Eq. (11) and raise the reheating temperature gradually. While the post-inflationary gravitino production increases proportionally to T R , the subsequentÑ 1 decay introduces the dilution factor which precisely compensates the previous effect. Thus, the number density of gravitinos is determined by an effective reheating temperature T R,ef f instead of the original T R . This effective temperature is given by However, if the reheating temperature rises above T R ≃ 10 12 GeV, the situation changes because now reheating takes place beforeÑ 1 oscillations start. While the initial gravitino production continues to grow with T R , the dilution factor remains constant, giving Applying the analysis of Ref. [23], we see that a gravitino of mass of order 1 TeV is consistent with a significantly extended range of post-inflationary reheating temperatures, T R < 10 13 GeV. Putting this upper bound together with our earlier lower bound fromÑ domination leads to an allowed range, 2 × 10 9 GeV < T R < 10 13 GeV, for successful sneutrino-dominated leptogenesis.

Conclusions
In summary, we have presented a higher-dimensional scenario, well motivated from a particle-physics perspective, in which cosmological heavy sneutrino domination occurs naturally and low-scale leptogenesis is responsible for the observed baryon asymmetry of the universe. The entropy produced in the decay of theÑ condensate dilutes unwanted relics from the period of reheating, alleviating in particular the danger of gravitino overproduction.
We briefly comment on density perturbations in this scenario. The dominant density perturbation can originate from the fluctuations ofÑ during inflation [25]. The deviation of the spectrum from scale invariance depends upon m 2 eff /H 2 inf , where m 2 eff is the effective (mass) 2 discussed at the beginning of Sect. 3. Scale invariance thus requires the dimensionless Kähler potential couplings between N and the inflaton to be slightly suppressed. Moreover, as in our scenarioÑ decay is the origin of baryon number asymmetry and dark matter, we necessarily have adiabatic perturbation dominance [26].
Thus, the role of the inflaton is reduced to providing a period of exponential expansion while its two main dynamical effects, the production of density perturbations and the reheating of the universe, are taken over by the heavy sneutrino. In fact, the presence of the inflaton mainly has a constraining effect -it has to decay sufficiently early to allow for heavy sneutrino domination and sufficiently late not to produce an excess of gravitinos. Thus, one might also wonder whether it is possible to get rid of the inflaton altogether. One obvious possibility would be to assume a sneutrino potential with a flat region away from the origin, so that a sufficiently long inflationary period driven by the sneutrino condensate is realized. It would be a very interesting and challenging task to understand the origin of such an unusual sneutrino potential.
Finally, we discuss a less radical way of avoiding the constraints associated with the decay of the inflaton entirely. If the inflaton potential is such that, in the true vacuum, the inflaton is massless, its energy density during the oscillation period decays faster 4 than that ofÑ, which varies with the scale factor R as R −3 . Thus, the desiredÑ domination is always obtained. The masslessness of the inflaton is technically natural since we do not require any non-gravitational coupling of the inflaton to the matter sector. It would be interesting to write down and analyse a well-motivated and complete inflation model with a potential that leads to such a 'harmless' late time behaviour of the inflaton.