Invisible decays of Higgs and other mesons in models with singlet neutrinos in large extra dimensions

In light of current atmospheric neutrino oscillation data, we revisit the invisible decay of the standard model Higgs boson and other pseudoscalar mesons which can be enhanced because of large number of KK modes in models with right handed singlet neutrinos in large extra dimensions. We find that the invisible decay rate of Higgs can be as large as $H\to b \bar b $ decay rate only for a very restricted region of parameter space. This parameter space is even further restricted if one demands that the dimensionless neutrino Yukawa coupling $\l$ is O(1). We have also studied the scenarios where singlet neutrino propagate in a sub-space, which lowers the string scale $M_{\ast}$ and keeps neutrino Yukawa coupling O(1). We have also considered decays of other spin-0 mesons to $\nu \bar\nu$ and found the rates to be too small for measurement.

The concept of large extra dimensions and TeV scale quantum gravity [1] introduced by ADD has attracted a lot of attentions recently. In this scenario, one has δ additional spatial dimensions of size R in which gravity propagates whereas the standard model particles with chiral fermionic content are confined to the usual 4-dimensions (4D). The effective 4D Planck scale, M P l ∼ 2.4 × 10 18 GeV , is related to the fundamental Planck scale M * in (4 + δ) dimension by Thus, for the large extra-dimensions, it is possible to have a fundamental scale M * as low as a TeV [1], helping to resolve the gauge hierarchy problem of the standard model. The size of the extra dimensions can be as large as ∼mm for δ = 2. Another interesting aspect of this scenario is the generation of a small neutrino mass [2,3]. The relatively small value of M * indicates the possibility of a seesaw mechanism, with right-handed (RH) singlet neutrinos also propagating in the full 4 + δ dimensions with gravity [4,5,6,7,8,9]. For illustration let us assume the case of a single extra-dimension labeled by y (x labels the usual 4D) [10].
A massless Dirac fermion N which is a standard model singlet lives in 5D. The Γ matrices in 5D can be written as The Dirac spinor N in the Weyl basis can be written as where ψ and χ are 2-component complex spinors (with mass dimension 2). The 5D kinetic term for N is and the interaction action is where, ℓ = (e, ν) is the standard model lepton doublet and H is the Higgs doublet and λ is a dimensionless Yukawa coupling in 5D. We have assigned N and ℓ opposite lepton number, so that the lepton number is conserved in Eq.(5).
In the effective 4D theory, N appears as a tower of Kaluza-Klein (KK) states: where ψ (n) are 4D states. Similarly, χ has a KK tower, χ (n) . Thus, in 4D, we have the effective action: From this expression it is clear that κ increases with m and M H , while it decreases with increase in δ and M * . Note that κ depends on m rather than m ν directly.
We now evaluate the ratio κ in light of the latest atmospheric neutrino oscillation data, which suggest ∆m 2 atm = (1.5 − 4.0) × 10 −3 eV 2 [12]. In our analysis we fix the highest neutrino mass which dominates H → νν rate to be m ν ∼ ∆m 2 atm and then solve for m from Eq. (10) in terms of M * and δ. The seesaw mechanism in Eq. (9) clearly favors hierarchical neutrino masses rather than degenerate mass spectrum ( which will need fine tuning of λ).
We consider two cases that respect the allowed range of ∆m 2 atm , (a) m 2 Another important consideration is the value of λ that ensues for a given m and M * . Clearly, the value of λ, which is the higher dimensional Yukawa coupling, should not exceed ∼ 10 − 20, otherwise we will be in a non perturbative regime. This is a serious constraint on allowed value of M * for a given δ.
In Figure 1  Similar variation is shown in Figure 1 (b), however, for m 2 ν ∼ 4.0 × 10 −3 . The situation is better in this case, since, higher value of m ν lead to larger value of m in the ratio κ. In this case also we have truncated each curve on the left by imposing the same condition as in Figure 1(a).
Before we proceed, we would like mention two points here. Firstly, beyond M H = 150 GeV, the dominant mode of Higgs decay is H → W W * , thus the H → bb branching ratio is very small, and any H → νν branching ratio will also become small. However, for the purpose of comparison we have presented the case of 200 GeV Higgs mass. One should also note that for the value of M * at which κ is O(1) and larger, corresponding Yukawa coupling λ is around 80 for δ = 3. Such a large value of λ may not be accepted from the perturbative point of view.
For a given value of m ν and M * , the value of λ can be obtained from Eq. (9) and Eq. (10): It is easy to show that if we limit λ ≤ 10 (to be in perturbative regime) , for m 2 ν = 1.5 × 10 −3 eV 2 we are restricted to M * ≥ 40 TeV and for m 2 ν = 4.0 × 10 −3 eV 2 , M * ≥ 64 TeV. The dependence on δ is very week on these bounds. Thus for such high values of M * the ratio κ is highly suppressed.
For completeness we discuss the case of degenerate neutrinos, separated by ∆m 2 consistent with the atmospheric and solar neutrino data. Recently WMAP [13] provided important new information on cosmic microwave background anisotropies. After combining the data from 2dF Galaxy Redshift Survey, CBI, and ACBAR [14], WMAP places stringent limits on the contributions of neutrinos to the energy density of the universe [15] for a single active neutrino, or m ν < 0.23 eV for three degenerate neutrinos. Using the value m ν = 0.23 eV and multiplying Eq. (13) by 3 for three families we can calculate κ. This is shown in Figure 2 . The general behavior of κ as a function of M * is similar to Figure 1.
The only point to be noted here is that, because of heavier neutrino mass, the allowed range of M * is much higher than the previous case. If we now impose the condition that λ ≤ 10 to satisfy the perturbative condition, M * becomes too heavy ∼ 200 TeV. For such a large value of M * , κ drops below 10 −4 for M H = 100 GeV and δ = 3, , and is essentially unmeasurable.
The situation is worse for higher values of δ.
In summary, we see from  (1), in that case, M * will be ≥ 100 TeV for m ν ∼ ∆m 2 atm and M * ≥ 200 TeV for m ν = 0.23 eV. This is perfectly fine, but for such a large value of M * , the invisible decay width of Higgs will be negligibly small compared to H → bb. The second choice suggested by [3,16] is to consider that the singlet neutrino propagates in a sub-space (δ ν ) of the full extra-dimension (δ) where gravity propagates. Assuming that all extra dimensions are of the same size R, in this case, the Dirac mass for the standard model neutrino now becomes Thus for δ ν = 5 and δ = 6 and with M * ∼ TeV, λ ∼ O (1), we obtain m 2 ∼ ∆m 2 atm . It has been shown by Agashe and Wu [16] that for the above choices of δ ν and δ, the constraints on M * from BR(µ → eγ) and π → eν, µν decays can be significantly weakened as compared to the minimal model, allowing the scale M * ∼ TeV. In this case, the maximum value of the physical neutrino mass for a given δ ν , δ and M * is given by [16]: Following Eq. (18), one can invert Eq. (11) We now compute the ratio κ as a function of M * keeping δ ν = 5 and δ = 6. In Figure 3 Figure 4 in the degenerate neutrino mass scenario. In this case, we find that κ can be larger than 1 for M H = 200 GeV. However, as mentioned before, this value of Higgs mass will not serve the purpose of looking for invisible decay modes of the Higgs boson. Hence, for practical purpose, we should look at values of κ for M H up to 150 GeV. It turns out that for M H = 150 GeV, κ ∼ 0.8 for M * ∼ 2.5 TeV, whereas, for M H = 100 GeV, κ can be at most 0.1.
In these two analysis, we have shown that only in a very small region of parameter space can the invisible decay of Higgs be as large as H → bb decay mode. The main restriction comes from the perturbative constraint on the Yukawa coupling λ.
For completeness we also discuss the case of asymmetric dimensions [5,8]. In this scenario neutrinos propagate in a sub-dimensional space of dimension δ ν of size R, whereas gravity propagates in space of dimension δ. The extra (δ − δ ν ) has a size r with (r << R).
In such a scenario Eq. (1) becomes In this case, the Dirac mass for the standard model neutrino becomes To satisfy the constraints from supernova 1987a, the scale 1/R > 10 KeV [7]. For such a value of 1/R the mixing angle (∼ mR/n) of any KK state with the standard model neutrinos becomes negligibly small and m ≈ m ν .
In this scenario, the ratio κ as defined in Eq.(13) turned out to be: To study the invisible Higgs decay in this scenario, we fix λ = 1 and take the same values of Higgs mass as before. We find κ = 0.17, 0.

Neutral Pion decays
We begin our analysis with the neutral pion. The effective Lagrangian for the process π 0 → ν Lν i R is in the standard model through Z exchange is : where, G F and F π are the Fermi coupling constant and pion decay constant respectively.
Using this effective Lagrangian we one determine the decay width of pion into ν Lν i R in the minimal model: We have computed the branching ratio BR(π 0 → ν Lν i R ) for m 2 ν = 4.0 × 10 −3 eV 2 and found the result is ∼ 10 −25 for δ = 3, M * = 200 TeV and λ ∼ 11. This predicted branching ratio is much smaller than the experimental upper limit 8.3 × 10 −7 at 90% C.L. [17].
The branching ratio can be higher by 4−5 order of magnitude for lower values of M * ∼ 45 TeV, but such a low value of M * correspond to unacceptably large λ ∼ 200. In the case of degenerate neutrinos, the above branching ratio does not change significantly, so we do not present any numerical results here.
Next, we compute the above decay widths in the sub-space scenario. In this scenario as shown earlier one can have λ of order one and also M * of about few TeV. With the following replacement δ → δ ν and (M P l /M * ) 2 → (M P l /M * ) 2(δν /δ) one can rewrite the decay width Γ (π 0 → ν Lν i R ) in sub-space model. In this case, we find that for m 2 ν = 4.0 × 10 −3 eV 2 , the BR(π 0 → ν Lν i R ) is of the order 10 −19 for M * ∼ 1.4 TeV corresponding to λ of O(1). The situation remain unchanged even with the assumption of degenerate neutrino masses. In the scenario, where the right-handed neutrinos propagate in the extra-dimensions with largest size, the BR(π 0 → ν Lν i R ) is too small to be observed.

B meson decays
In the standard model, the process B → νν receives contributions from Z-penguin and box diagrams, where the dominant contribution comes from intermediate top quark loop.
Off-shell Z and W exchanges have contributions from would be Goldstone modes that couple to right handed neutrinos. The effective Lagrangian for B → νν for each neutrino is given by is the Fermi coupling constants, α is the fine structure constant ( at the Z mass scale), F B is the B-meson decay constant, θ W is the weak angle and V * td V tb are products of CKM matrix elements. The Wilson coefficient C SM 11 at the leading order is given by: In models of large extra-dimension, B → νν gets additional contribution from Higgs exchange contribution. The effective flavor changing vertex (bdH 0 )can be obtained from [18] where, V ij are the elements of the Kobayashi-Maskawa matrix and m i are the corresponding quark masses flowing in the loop.
The effective Lagrangian for the process B → νν retaining only the top quark contribution is: Even in the sub-space scenario, the branching ratio does not get any significant enhancement, irrespective of the different neutrino masses, assumptions considered previously. In the scenario, where the right-handed neutrinos propagate in the extra-dimensions with largest size, the BR(B → ν Lν i R ) is too small to be observed.

Conclusions
In this analysis we have studied the possible enhancement of invisible decay widths of the standard model Higgs boson and other pseudoscalar mesons in the model of singlet neutrinos in extra-dimensions. In the case of Higgs boson decay we have found that in certain range of extra-dimension parameter space, the branching ratio of Higgs into invisible mode can be ≥ BR(H → bb). Unfortunately, the higher dimensional Yukawa coupling λ takes on large (≥ 100) values in that parameter space. For λ ≤ 10, the H → νν rate is a tiny fraction of the H → bb rate.
We have also studied the invisible decay rate of Higgs in the scenario, where right-handed neutrinos are in sub-space (δ ν < δ), which is the modification of the minimal model, required to keep λ ≤ O(1). It has been shown that to have a consistent model allowed by different experimental constraint , one should have δ ν = 5 and δ = 6. In this scenario, the invisible decay rate of Higgs can compete to that of H → bb, though for a very small range of We have also studied the decay rates π 0 → ν Lν i R and B → ν Lν i R in all these scenarios. Unfortunately in both of these decays the new effects are negligibly small to be measured.