U(1)B−L extra-natural inflation with Standard Model on a brane

The interrelation between inflationary cosmology and new physics beyond the Standard Model (SM) is studied in a U(1)B−L extension of the SM embedded in a (4+1)-dimensional spacetime. In the scenario we study, the inflaton arises from the Wilson loop of the U(1)B−L gauge group winding an extra-dimensional cycle. Particular attention is paid to the coupling between the inflaton and SM particles that are confined on a brane localised in the extra dimension. We find that the inflaton decay channels are rather restricted in this scenario and the resulting reheating temperature is relatively low. ar X iv :1 31 0. 46 46 v1 [ he pph ] 1 7 O ct 2 01 3


Introduction
The precision of the Cosmic Microwave Background (CMB) anisotropy observations has started to rule out some of the inflation models [1]. However, CMB data alone still accommodates a large class of them. In order to narrow down further likely candidates, it is useful to study possible relevance of the inflaton to physics in other eras. In particular, at the time of reheating, the inflaton decays to Standard Model (SM) particles so that the standard hot big bang can proceeed, the nature of the interaction between the inflaton and the SM is thus crucial.
Large field inflation models had attracted attention because of the possibile detection of tensor modes in CMB polarization in the near future [2]. It is theoretically challenging to construct natural large field inflation models, since effective field theory approach usually breaks down in these models. Extra-natural inflation [3,4], which is based on a gauge theory in higher-dimensional spacetime, is one way to circumvent this difficulty by using non-local operator (Wilson loop) in the extra dimension.
It is an interesting question what should be the gauge group for extra-natural inflation. As we review in the next section, it turns out that to explain the CMB data the gauge coupling for extra-natural inflation must be very small [3,4]. This makes it difficult to identify the SM gauge groups as that for extra-natural inflation, as their couplings at the electro-weak scale are orders of magnitudes larger than that required for extra-natural inflation. Therefore we shall look for other gauge groups in models beyond the SM (BSM).
Gauged U (1) B−L extension of the SM [5,6,7,8] is ubiquitous in scenarios of BSM physics. A nice feature of it is that the existence of right-handed neutrinos is made natural by the necessity of gauge anomaly cancellation. It also makes R-parity exact in supersymmetric versions of the SM, and it appears as an intermediate stage in the symmetry breaking pattern of grand-unified models down to the SM, as well as in higherdimensional embeddings of the SM in string theory constructions. Apart from the formal theoretical considerations, phenomenologically, having a new gauge boson and scalars neutral under the SM gauge group can give rise to novel effects observable in future collider experiments.
In this letter, we study extra-natural inflation with U (1) B−L as the gauge group. In the scenario we study, the bulk spacetime is (4+1)-dimensional with the extra dimension compactified on a circle, SM is confined on a (3+1)-dimensional brane localised in the extra dimension, and the inflation arises from the Wilson loop of the U (1) B−L gauge field living in the full (4+1)-dimensional bulk. In the following, we explore the interrelation between inflationary cosmology and particle physics in this setting. 1 The rest of the letter is organised as follows. The relevant ingredients of extra-natural inflation is reviewed in Sec. 2. The details of our U (1) B−L extension of the SM is discussed in Sec. 3. The decay of the inflaton to SM particles is studied in Sec. 4. We end with a summary and discussions in Sec. 5.

U (1) B−L extra-natural inflation
Extra-natural inflation [3,4] is a version of natural inflation [11] whose typical potential takes the form where φ is the inflaton which, in extra-natural inflation, is the zero-mode of the fifth component of some bulk gauge field. In the scenario we study here, it is that of the U (1) B−L gauge group. From (2.1) the slow-roll parameters are given by Here denotes derivative with respect to φ. The slow-roll conditions amount to In extra-natural inflation, f and V 0 are estimated as [3] f = 1 and (2.6) Here, g 4 is the (effective) four-dimensional gauge coupling, and L 5 is the radius of the compactified fifth dimension. The constant c 0 is determined by the matter content in the bulk, with the relevant ones being fields charged under the gauge symmetry of interest and whose masses are below or of the order of 1/L 5 [12]; each of these field makes an O(1) contribution to c 0 . 2 In order for quantum gravity corrections to be small, we need where M 5 is the five-dimensional (reduced) Planck scale, which is related to the fourdimensional reduced Planck scale M P 2.4 × 10 18 GeV by Thus from (2.5) Since f is directly related to the CMB observations, and g 4 is a basic parameter in the U (1) B−L extension of the SM, we shall take f and g 4 as the independent parameters, and regard L 5 and M 5 as functions of them. It is convenient to introduce a dimensionless parameter which measures the strength of quantum gravity corrections; (2.7) then amounts to 3 5 1. Although 5 is not an independent parameter, it is sometimes convenient to use 5 instead of g 4 . In terms of 5 and f , g 4 is expressed as The number of e-folds as a function of φ is given by Here, φ e is the value of the inflaton field at the end of inflation defined by V (φ e ) = 1, where the slow-roll condition (2.4) breaks fown. 3 This gives and plugging (2.13) into (2.12) we obtain (2.14) In slow-roll inflation, the tensor-to-scalar ratio, r, and the spectral index, n s , are given by The scalar-to-tensor ratio and the spectral index estimated from various combinations of the Planck data and other observations give at 95% CL: r 0.12 and 0.94 n s 0.98 at the pivot scale k * = 0.002 Mpc −1 [1]. Below, except for r and n s whose value we take always at the pivot scale, we shall use the subscript * to indicate that the value is taken at the pivot scale.
As can be seen from (2.2), (2.3) and (2.14), r and n s only depend on f and N * in extra-natural inflation, and so constraints on r and n s constrain f for a given N * . We plot the dependence of r and n s on f at fixed values of N * in Figs 1 and 2, respectively. We see that for N * = 50, we have f 10M P from r 0.12 and f 5M P from n s 0.94. We will see later when considering the inflaton decay that N * 50 is natural for the scenario we study here.  The power spectrum of the slow-roll inflation is given by (2.16) This should be compared with the observed value P ζ (k * ) = 2.2 × 10 −9 [1]. It determines the Hubble scale, H * , when the pivot scale exited the horizon, and thus the energy density at that time, ρ * 3M 2 P H 2 * , as a function of f and N * . Its dependence on f and N * is mild, and we obtain ρ * 10 16 GeV, see Fig. 3. On the other hand, from the Friedman equation for spatially flat Universe in the slow-roll approximation, we obtain .
In the last line we have made it explicit that φ and V are functions of f and N . Thus given 5 , f and N * , c 0 is determined from the observed value P ζ (k * ) = 2.2 × 10 −9 by (2.18). The behaviour of c 0 as a function of 5 is plotted in Fig. 4. We observe that c 0 grows as 6 5 . Also, c 0 grows rapidly with f , as seen in Fig. 5. Since each field charged under U (1) B−L with mass L −1 5 makes an O(1) contribution to c 0 , if it is much larger than unity it may not be natural. 4 Therefore we regard smaller values of 5 and f , viz.

U (1) B−L extension of the Standard Model
There are several possibilities for the U (1) B−L extension of the SM, particularly with regards to the charge assignment of the scalar field that would break the U (1) B−L symmetry. Table 1 lists the particle content and the charge assignments of the particular U (1) B−L extension of the SM we consider here. In our set-up, we envisage all the SM particles and the right-handed neutrinos living on a four-dimensional brane, while the U (1) B−L gauge fields, A M , and a complex scalar, Σ, responsible for the eventual U (1) B−L breaking living in the five-dimensional bulk. In string theory this set-up may be realized, for example, when the SM fields and the right-handed neutrinos live on a (3+1)-dimensional D-brane localised in the extra dimension, while the bulk fields arise from higher dimensional Dbranes.
The potential for the scalar sector renormalizable in four dimensions is given by Here, Σ 0 is the zero-mode of Σ in the fifth direction. After spontaneous symmetry break- where h and s are excitations about the minimum, which is given by Note that the W boson mass fixes v H = 246 GeV. In terms of h and s, the quadratic part of the potential is given by where is the tree-level mass-squared matrix for h and s, and we have used the minimization condition for the potential. Diagonalizing, the physical mass eigenstates are defined by with the mixing angle given by The masses of the physical states are then given by For |λ 3 | 1 and |v H /v Σ | 1, we can expand the square root and obtain Assuming no coupling between the Higgs H and the scalar Σ 0 at tree level, the mixing term H † HΣ * 0 Σ 0 is induced at one-loop level [13] through interactions with neutrinos responsible for the seesaw mechanism [14,15,16,17]: mixing term contributes to the Higgs mass is estimated as where we have used the seesaw formula m ν ∼ Y 2 2 being the mass of the right-handed neutrino. Given the observation of the Higgs boson with mass 126 GeV at the LHC [18,19,20], we should have |δm 2 H | 100 GeV if naturalness is a criterion. Thus if we take m ν ∼ 0.1 eV, we have M N 10 7 GeV from (3.11) and hence v Σ 10 7 /Y N GeV. This translates to an upper bound on the mixing coupling Assuming Y N O(1), the mass of the physical U (1) B−L gauge boson is estimated as From collider experiments, one has m Z ≥ g 4 × (6 TeV) for a U (1) B−L Z boson [21]. Since g 4 10 −3 , there are no stringent bounds on m Z .

The inflaton decay
The coupling between the inflaton and the SM particles is crucial at the time of reheating. Let us first consider the following Z 2 transformation: We choose the origin of the x 5 coordinate to be where the brane is localised. We assume there are no other fields with Z 2 -odd charges under (4.1) that are lighter than A 5 . Then if this Z 2 transformation is an exact symmetry, the inflaton is absolutely stable. This will be a problem, however, since then the Universe could not be heated to bring forth the standard hot big bang cosmology. We therefore introduce a five-dimensional Chern-Simons term, which breaks the Z 2 symmetry: where A = A M dx M , F = 1 2 F M N dx M dx N , and k is some integer. Here, A M is the U (1) B−L gauge field with mass dimension one, which is related to the canonically normalized fields by where A M is the U (1) B−L gauge field canonically normalized in five dimensions, A µ that in four dimensions, and A M 0 the zero-mode of A M in five dimensions.
The four-dimensional interaction of the zero-modes following from (4.2) is Here the subscript 0 denotes that they are (made from) zero-modes in the fifth direction.
The coupling (4.5) gives the dominant contribution to the decay width at the tree level: where m φ is the mass of the inflaton. As we have seen, c 0 is determined by (2.18) once f and g 4 are given. This then determines m φ : (4.7) The U (1) B−L gauge bosons decay to SM particles via the minimal couplings. As this proceeds much faster than the inflaton decay, the reheating temperature is governed by the inflaton decay width (4.6). It is estimated as where in the last line, we have used the preferred values f 5M P and 5 5, which gives m φ 10 13 GeV. The factor g (T ) is the effective relativistic degrees of freedom at temperature T . For T R 1 ∼ 10 GeV, g (T R ) 60 ∼ 90. From (4.8), the reheating temperature is much smaller than the U (1) B−L breaking scale given by v Σ O(10 7 ) GeV, when k is O(1 − 10). Comparing (4.8) with the standard estimate of the number of efolds [22]: we observe that N * 50 is natural in our model, as advertised earlier.

Summary and Discussions
In this letter, we have studied the interrelation between cosmology and particle physics in U (1) B−L extra-natural inflation with a gauged U (1) B−L extension of the SM localised on a brane. The cosmological observation constrains the value of the U (1) B−L gauge coupling to g 4 10 −3 , which in turn constrains the particle physics scenario at high energy assuming naturalness. On the other hand, with SM particles localised on a brane, allowed interaction between the inflaton and the SM particles are restricted. Together with the value of g 4 , the decay width of the inflaton and the reheating temperature are determined.
By tuning of a few parameters or with some slight extension, our model may also be able to explain other cosmological observations such as the Baryon number asymmetry of the Universe and the dark matter abundance. Indeed, the right-handed neutrinos could play a role in the former through the leptogenesis. They are also dark matter candidates. Another possible dark matter candidate, which may be included in our model, is a light scalar field odd under the reflection of the extra dimension (4.1). These merit further investigations.
Our main purpose in this letter is to present an example in which the relation between the BSM physics and the inflation physics are specified, and theoretical and observational constraints on one side constrains the other. We discussed one example here, but there can be several other possibilities, even within gauged U (1) B−L extensions of the SM. For instance, one may put some of the SM fields in the bulk. It will be interesting to explore those related scenarios.