Constraining Proton Lifetime in SO(10) with Stabilized Doublet-Triplet Splitting

We present a class of realistic unified models based on supersymmetric SO(10) wherein issues related to natural doublet-triplet (DT) splitting are fully resolved. Using a minimal set of low dimensional Higgs fields which includes a single adjoint, we show that the Dimopoulos--Wilzcek mechanism for DT splitting can be made stable in the presence of all higher order operators without having pseudo-Goldstone bosons and flat directions. The \mu term of order TeV is found to be naturally induced. A Z_2-assisted anomalous U(1)_A gauge symmetry plays a crucial role in achieving these results. The threshold corrections to alpha_3(M_Z), somewhat surprisingly, are found to be controlled by only a few effective parameters. This leads to a very predictive scenario for proton decay. As a novel feature, we find an interesting correlation between the d=6 (p\to e^+\pi^0) and d=5 (p\to \nu-bar K+) decay amplitudes which allows us to derive a constrained upper limit on the inverse rate of the e^+\pi^0 mode. Our results show that both modes should be observed with an improvement in the current sensitivity by about a factor of five to ten.


Introduction
Although yet to be seen, proton decay is an indispensable tool to probe nature at truly high energies (∼ 10 16 GeV). It still remains as the missing piece of grand unification [1,2,3]. In the light of a new set of planned detectors including those at the forthcoming deep underground laboratory DUSEL [4] and the HyperKamiokande, we propose to address in this paper certain well known but partially unresolved theoretical issues of supersymmetric (SUSY) grand unification (GUT) which are especially relevant to proton decay.
Strong empirical support for grand unification arises not only from the observed quantum numbers of quarks and leptons and the quantization of electric charge, but in particular from the meeting of the three gauge couplings at a scale M GUT ≈ 2·10 16 GeV that occurs in the context of low energy SUSY [5], and the tiny neutrino masses, as observed in neutrino oscillation experiments. The latter fit extremely well with GUT symmetries that include the symmetry SU(2) L × SU(2) R × SU(4) C [1], the minimal such symmetry being SO(10) [6]. We will therefore discuss proton decay in the context of supersymmetric SO (10). The purpose of the present paper is to pay special attention to the problem of the so-called doublet-triplet (DT) splitting and to study the implications of its resolution for proton decay.
The DT splitting problem is common to all grand unified theories based on simple gauge groups.
In SUSY SO(10) models the two Higgs doublets of MSSM, a color triplet and an anti-triplet lie (typically) in a 10-dimensional representation H (10). The color triplets need to be superheavy so as to avoid rapid proton decay and also to preserve gauge coupling unification. Keeping the doublets light and the triplets superheavy self-consistently is the doublet-triplet splitting problem.
A natural solution to this problem, avoiding severe fine-tuning is realized in SUSY SO(10) by the so called Dimopoulos-Wilczek (or the missing VEV) mechanism [7]. It involves a coupling of This structure contributes to the triplet and not to the doublet masses, and thereby can lead to natural DT splitting without fine-tuning.
Given the very large hierarchy between the doublet and triplet masses, however, one must ensure: (i) that the missing VEV pattern for A(45) in Eq. (1) is stable to a high enough accuracy in the presence of all allowed higher dimensional operators; (ii) that there are no undesirable pseudo-Goldstone bosons; and (iii) that there are no flat directions which would lead to VEVs of fields undetermined. Furthermore, (iv) one must also examine, by including all GUT-scale threshold corrections to the gauge couplings, the implication of the doublet-triplet splitting on coupling unification and on proton decay. To our knowledge, while some of these issues have been partially addressed in the literature (e.g. see [8], [9], [10]), and major progress was made in Ref. [11] with regard to the issues (i) and (ii), simultaneous resolution of all four issues has so far remained a challenge.
In this paper we present a predictive class of SO(10) models, based on a minimal Higgs system, in which all the issues of DT splitting mentioned above are resolved, and where the threshold corrections to the gauge couplings and their implications for proton decay are properly studied as well. The Higgs sector we consider has a single adjoint, along with vectors and spinors. Such a low dimensional Higgs system would lead to smaller threshold effects [8,10], unlike in models [8,10] which employ multiple adjoints and/or 54 dimensional Higgs. 4 A postulated Z 2 -assisted anomalous U(1) A symmetry (which may have a string origin [13]) plays a crucial role in obtaining our results. We find somewhat surprisingly that the GUT scale threshold corrections to α 3 (M Z ) are determined in terms of a very few parameters. This makes the model rather predictive for proton decay. As a novel feature, we find an intriguing correlation between the d = 6 and d = 5 proton decays, which respectively lead to p → e + π 0 and p → νK + as the dominant decay modes. The correlation is such that the empirical lower limit on Γ −1 (p → νK + ) provides a constrained upper limit on Γ −1 (p → e + π 0 ). Our results show that both decay modes should in fact be discovered with an improvement in the current limits on lifetimes by about a factor of five to ten. 4 An alternative class of SO(10) models utilizing larger dimensional (e.g. 126) Higgs fields has been studied in Ref. [12]. These models have the interesting feature that R-parity is automatic, being part of the gauge symmetry.
However, threshold corrections are rather large in these models, making quantitative predictions for α 3 (m Z ) and proton decay difficult (see attempts in this regard by Aulukh and Garg [12]).

Stabilizing doublet-triplet splitting
In order to break SUSY SO (10) to the supersymmetric standard model with a stabilized DT sector, and for the subsequent breaking of the electro-weak symmetry, we shall use a minimal low  (10), as well as two SO(10) singlets S and Z. The second spinorial pair C ′ +C ′ is introduced, following Ref. [11], to avoid pseudo-Goldstone degrees of freedom while maintaining the Dimopoulos-Wilczek VEV structure for A (cf: Eq. (1)). The S and Z superfields are needed to fix various VEVs in the required directions through their superpotential couplings.
We supplement the gauge symmetry by a Z 2 -assisted anomalous U(1) A symmetry in order to stabilize the VEV pattern of Eq. (1) [14], [10]. The charges of the Higgs fields and those of the three matter families 16 i under U(1) A × Z 2 are listed in Table 1. Here k is a positive integer which is unspecified for the moment. The superpotential of the symmetry breaking sector, consistent with these symmetries, is For simplicity we assume that the SO(10) contractions in the CC terms with coefficients b 1,2 in Eq.
(3) are in the singlet channel. In the second term of Eq. (4) the operator Z k can appear only when k is even. While our consideration of DT splitting will hold for all k, if k is odd, matter parity is automatic, being part of U(1) A . The choice of k = 5, which we will use, is phenomenologically preferred, in particular for suppressing adequately all d = 5 proton decay operators including those induced by Planck scale physics. Higher order operators such as A 6 /M 3 * etc. are not exhibited in Eqs. (2)-(4) because they are inconsequential for our purposes. The charges q 1,2 and the parities P 1,2 of the first two families are left unspecified for the present. They will however be relevant for the generation of quark and lepton masses.
Typically, we expect that the non-renormalizable operators such as λ A and λ ′ A -terms would be induced by quantum gravity effects involving exchange of heavy states in the string tower. Thus, we expect the cut-off scale M * ∼ M Pl or M String ∼ 10 18 GeV. We shall take all dimensionless couplings to be of order unity, i.e., in the range (1/4 − 2).
Using the SUSY preserving condition F Z = F S = 0, together with the choice C = C = c, A = 0 (which is one allowed option among the discrete set of degenerate vacuum solutions), we get CC ′ = C C ′ = 0 and C ′ = C ′ = 0. The VEV of A is then determined entirely by W 1 of Eq. (2). Setting F A = 0, we find a solution in the B − L direction as in Eq. (1), with With λ A , λ ′ A ∼ 1 and M * ∼ 10 18 GeV, we need to choose M A ∼ 10 15 GeV to obtain a ∼ M GUT ≈ 2 · 10 16 GeV. Demanding F -flatness conditions F C ′ = FC′ = 0 and using the notations z = Z and We note that for all dimensionless couplings in the Lagrangian being in the range (1/4 − 2), the effective couplings ρ 1,2 can naturally take values as small as about 1/50. The sum of the VEVs gets further constrained as follows. The anomalous U(1) A symmetry, presumed to have a string origin, generates the Fayet-Iliopoulos term ξ through quantum gravity, which is given by [13] where g st denotes the string coupling and M Pl ≃ 2.4·10 18 GeV is the reduced Planck mass. In our model, the particle spectrum of Table 1 would lead to Tr(Q A ) = −8 − 60/k + 16(q 1 + q 2 ) = −84/5, (for k = 5, q 1,2 = −1/2 + 3/k, see later). This value will however be modified if there are additional singlets in the full theory. (Semi-realistic string solutions [15], possessing an anomalous U(1) A , typically lead to |TrQ A | ≈ 30 − 100.) With the charges in Table 1, the vanishing of Thus, the VEVs of all the fields get determined. We see that quite naturally, the VEVs c, z ∼ (few−10)×M GUT , and s ∼ (10 −2 −10 −1 )×M GUT can arise, with the precise values depending on the order one couplings. Let us note that this setup also allows for additional singlet fields {P i } which can play a role in the D A = 0 condition (for P i with positive U(1) A charges) and can modify these estimates somewhat, without upsetting the stability of DT splitting.
Substituting the VEVs of the heavy fields in Eqs. (3) and (4), we derive the mass matrices M D and M T for the SU(2) L doublets and SU(3) c -color triplets (written in the SU(5) notation): can not couple to H 2 , which is also positively charged. Thus, with η D = 0 one pair of the Higgs doublets will be massless, while the remaining three pairs of doublets become superheavy. The role of the Z 2 symmetry is that it allows the coupling of H to H ′ only through A (or odd powers of A).
Such couplings, however, do not generate a doublet mass due to the VEV structure in Eq. (1) of A . The VEV pattern of A along the B − L direction is also guaranteed to be stable because of the U(1) A symmetry. Indeed, note that the symmetry U(1) A does not allow any superpotential coupling involving A, C andC of the form A n (CC) m . It is only these couplings which, if allowed, would have upset the missing VEV pattern of Eq. (1). Their absence to all orders thus guarantees that the pattern of Eq. (1) is absolutely stable (barring of course SUSY breaking at the TeV scale which is safe). As far as the color-triplets are concerned, since η T = 0 in Eq. (6) for the triplets, all four pairs become super-heavy, just as desired.
The two massless Higgs doublets which emerge from Eq. (6) The angle γ is determined in terms of the parameters of the superpotential. It is related to the MSSM parameter tan β as tan β = mt m b cos γ. Note that, unlike in many SO(10) models, the MSSM parameter tan β is not required to be large here. It would turn out that conservative upper limits on proton lifetime would correspond to smaller values of tan β.
The µ-term, the coefficient of h u h d term of MSSM superpotential, is generated within our model in a simple way. In the unbroken SUSY limit, µ-term is zero since terms such as H 2 are forbidden.
After SUSY breaking, the adjoint A(45) develops a VEV∼ m susy along its I 3R direction, correcting the zeros of Eq. (1), which generates the µ-term. This occurs since the inclusion of the soft SUSY breaking terms induces VEVs ∼ m susy for the fields C ′ andC ′ along their ν c -like scalar components. These will trigger the VEV (∼ m susy ) of A(45) in the I 3R direction. From Eq. (4) we obtain λ 1 HAH ′ → m susy h u h d , and thus µ ∼ m susy ∼ TeV, independent of the integer k. Thus the present setup provides a simple and elegant solution to the µ problem without any new ingredients.
yields the mass of the color octet and SU(2) L triplet in A(45): M Σ ≡ M 8 = 2M 3 . We see from Eq. (9) that two pairs of (u c , q, e c )-like states are massive. The third massless pair (10 + 10 of SU (5)) is eaten by the corresponding massive gauge superfields of SO(10)/SU (5). Denoting the masses of the components of (10 + 10)'s by curly symbols (e.g. U c 1 ≡ M(u c 1 ) etc.), we derive from Eq. (9): These expressions will be useful for the computation of threshold corrections in the model.
The masses of the heavy gauge boson superfields corresponding to the broken generators of SO(10) are given by (see e.g., the second paper in Ref. [8] and [9]): where g is the unified gauge coupling at the GUT scale. Given the p andp dependence of the masses given above, we see that, except for the e c -like states, threshold corrections from all other states in the (10 + 10) matter sector cancel precisely against those from the corresponding states in the gauge sector. 5 Details of estimating the VEVs based on explicit solutions to the D A = 0 condition will be presented in a forthcoming longer paper [16].
An accidental N = 4 supersymmetry present in the model (the gauge bosons and three pairs of matter fields in the (10 + 10) sector form an N = 4 SUSY gauge multiplet) is responsible for this cancelation. This results in an enormous reduction of the parameters, rendering the model very predictive for proton decay.
We have presented the whole spectrum of the theory, except for the singlet sector, which is not relevant for the calculation of threshold corrections. We have however, analyzed the singlet sector and verified that there are no unwanted pseudo-Goldstone states in the model. While it might appear that there is a U(1) symmetry associated with the "integer" k in Table 1, it turns out that this is a linear combination of U(1) A and B − L, and its breaking does not lead to a pseudo-Goldstone boson.
The evolution of the three gauge couplings in the model with momentum is shown in Fig. 1, which takes into account all the threshold effects. It is clear from Fig. 1 that the three couplings merge into one at a unification scale M GUT ∼ 10 16 GeV. Furthermore, we see that the unified SO(10) gauge coupling remains perturbative to scales well above M GUT . This is a desirable feature which not all SO(10) models have.
3 Novel correlation between d = 5 and d = 6 proton decays Writing down the three RG equations for α −1 1,2,3 and eliminating the unified gauge coupling α G we obtain 1, w ) −ln We have taken GUT scale threshold corrections in one loop approximation. The quantities ∆ (2) i, w include weak scale threshold corrections and 2-loop running effects for the gauge couplings, including Yukawa interactions. Their values depend on the SUSY particle spectrum. We carry out our analysis within the minimal N = 1 SUGRA scenario [17] with family universal parameters. While we vary these parameters to draw our conclusions, for concreteness, we consider the set of values:  (5)).
It is important that the ratio r ≡ M Σ /M X entering into Eq. (11) is constrained by symmetries of the model. Using expressions for M Σ , M A and M X presented earlier, we obtain: The range for r is obtained by noting that λ ′ A is allowed by symmetries of the model and thus naturally expected to be of order one. We have thus taken, 1/5 < ∼ λ ′ A < ∼ 2 (say), and have set g 2 ≃ g 2 GUT ≈ 0.63(1 ± 0.10), M X ≈ (0.6 − 1) × 10 16 GeV (see discussion after Eq. (14)), while M * ≈ M Pl ≃ 2 × 10 18 GeV. This restriction on r will be an important ingredient in the derivation of an upper limit on Γ −1 (p → e + π 0 ). Eliminating p/p from Eqs. (10) and (11) we obtain a correlation between M eff and M X for a given r: where η γ ≃ 0.6 accounts for the running of cos γ, and δα −1 and d = 6 (p → e + π 0 ) decay amplitudes, Eq. (13) in turn provides a correlation between the rates of these two otherwise unrelated decay modes. Such a correlation exists in minimal SUSY SU(5) as well [18], but that leads to predictions for α 3 (M Z ) and d = 5 proton decay rate which are inconsistent with experiments [19]. Exceptions to this conclusion has been suggested in Ref. [20] which uses higher dimensional operators. This however leads to large threshold corrections, making the apparent unification of gauge couplings with low energy SUSY somewhat coincidental.
For review of proton decay in SU(5) and in alternative scenarios see Ref. [21]. Now, using expressions for proton decay rates (see below) one finds that the empirical lower limit on Γ −1 (p →νK + ) requires that M eff > ∼ 2.91 · 10 19 GeV (for reasonable scenarios for the Yukawa couplings, see discussions in Sec. 4.1), while that on Γ −1 (p → e + π 0 ) requires (owing to Eq. (13)) r < ∼ 1/150. Using the particular choice of SUSY parameters stated above, and the ranges for M eff and r as given in Eqs. (8) and (12) consistent with experiments, the d = 5 proton decay rate is in full accord with the experimental limit.

Nucleon decay
There are two main mechanisms for proton decay corresponding to d = 5 and d = 6, which respectively yield p →νK + and p → e + π 0 as the dominant decay modes. Although apriori these two modes are largely independent, owing to the correlation given in Eq. (13) and Fig. 2, they get linked in our model such that the observed lower limit on the inverse decay rate of either mode implies an upper limit on that of the other. The latter is found to be especially constrained for the p → e + π 0 mode. The rates for d = 6 decay modes p → e + π 0 and p →νπ + , which are largely independent of the details of Yukawa couplings and SUSY spectrum, are given by: where f (p) = 4 + (1 + 1/(1 + p 2 )) 2 . Here α H denotes the hadronic matrix element. Recent lattice calculation yields α H ≃ 0.012 GeV 3 at µ = 2 GeV [22]. D and F are chiral lagrangian parameters with D ≃ 0.8, F ≃ 0.47. g X denotes the effective X, Y boson coupling at M X .
The correlation curves (see Fig. 2 for a representative case) restrict M X in the range of about (6 − 10) × 10 15 GeV. Taking an average of the three gauge couplings, which nearly unify at M X , lying in the range as given above (see Fig. 1), we obtain α G (M X ) = g 2 X /(4π) ≃ (1/20)(1 ± 0.1), where the error reflects variations in the GUT scale spectrum or equivalently in the parameters of the superpotential lying in a natural range. The function f (p) varies between the limits 8 and 5 as p varies from 0 to ∞; correspondingly one obtains Γ(p → e + π 0 )/Γ(p →νπ + ) ≃ (1, 1.4, 2.5) for . The case of p → ∞ represents the SU(5) limit. If this branching ratio is measured to be significantly smaller than 2.5, that would be strongly suggestive of SO(10) (as opposed to SU(5)) grand unification. The quantity A R in Eq.
Let us now briefly discuss the neutrino sector. The relevant operators, responsible for generating heavy Majorana masses for the right-handed neutrinos are: Here, M N , M N ′ and M N ′′ represent masses of additional singlets which turn out to lie in the range of (few -100)M GUT [16]. We assume that in the first of these couplings the The heavy Majorana mass matrix M R is given by: The Dirac mass matrix M νD at GUT scale can be obtained from M u (see Eq. (16)) by the replacement ǫ ′ → −3ǫ ′ . We can take the two dimensionless parameters (a, b) and the mass parameter M 0 as input to fix ∆m 2 atm , θ 12 and θ 23 . Two observables, viz., ∆m 2 sol /∆m 2 atm ≃ m 2 /m 3 and θ 13 , will then be predictions of the model. The structures given in Eq. (19) are valid at GUT scale. Applying renormalization, including threshold effects due to the different ν c masses, with θ 12 ≃ 30 o and θ 23 ≃ 43 o as inputs, we obtain m 2 /m 3 ≃ 0.13 and θ 13 ≃ 3.6 o as predictions. Such a fit is realized by choosing a = 0.0252e −0.018i , b = 1.61 · 10 −6 e −1.592i , and M 0 = 1.89 · 10 13 GeV. The corresponding ν c masses are (M R1 , M R2 , M R3 ) = (3.04 × 10 7 , 1.2 × 10 10 , 1.79 × 10 13 ) GeV. These results include all the relevant RG running effects. One sees broad, although not precise, agreement with data. We consider this fit, which provides large neutrino oscillation angles, together with small quark mixing angles as well as observed CP violation as fairly successful and highly nontrivial, especially in a quark-lepton unified framework with a stabilized doublet-triplet splitting.
With the Yukawa couplings specified, the inverse of the sum of partial decay widths, in p →νK + , is computed to be: Here K ν d=5 denotes a sum of contributions to the total decay rate from the three neutrino flavors, reflecting the dependence of the d = 5 operator on the Yukawa couplings: K ν d=5 ≡ |A νe | 2 + |A νµ | 2 + |A ντ | 2 . Each individual A ν i receives contributions from three types of diagrams leading to the d = 5 operator: (a) those with only the first two families in the external legs, (b) those having the quark doublet of the third family in just one external line, and (c) those having the same as in (b) in two external lines. The last two contributions incorporate the short distance renormalization of the d = 5 operator that arises through the running of the top quark Yukawa coupling, from the GUT scale to the weak scale. Contributions from all three diagrams are found to be important, especially for |A νµ | and |A ντ |. The net result is that |A νe | ∼ O(10 −1 ), |A νµ | ∼ |A ντ | ∼ O(1) and K ν d=5 ≃ 3.1 [16].Ā α S in Eq. (20) denotes the short distance RGE factor for the d = 5 operator, corresponding to the running from the GUT scale to the weak scale, that arises purely from the gauge interactions, without the effects of the top quark Yukawa coupling. Note thatĀ α S defined here differs from the RGE factor A S defined conventionally [18] in that A S includes the effect of the running of m c m s in going from low energies to the GUT scale,Ā α S does not.Thus, A S =Ā α S J, where J = (m c m s ) GUT /(m c m s ) µ ∼ O(10 −1 ) for µ = 2 GeV (with low tan β ∼ 3 to 10).
We can now discuss the derivation of an upper limit on M X and thus on Γ −1 (p → e + π 0 ). Owing to Eq. (13) this would correspond to the minimum allowed value of M eff . Now, taking conservatively mq < ∼ 1.5 TeV and the experimental lower limit mW > ∼ 125 GeV, the observed lower limit on Γ −1 (p →νK + ) > ∼ 2.8 × 10 33 yrs [23] yields (via Eq. (20)): (M eff ) min > ∼ 2.91 × 10 19 GeV. This in turn yields by using Eq. (13) (14)) would be 1.61×10 34 yrs. If the uncertainties in all these parameters are stretched to their extremes, each in a direction so as to prolong the lifetime, the stated upper limit could increase by atmost a factor of 10.8. Considering that all the uncertainties having such extreme values, and in the same direction, to be very unlikely, we would regard something like the geometric mean of the two upper limits corresponding to the central and extreme values of the parameters to be a more realistic, yet conservative, upper limit for the lifetime. We thus predict: If mq < 1.5 TeV, or mW > 125 GeV, or r > 1/300, or tan β > 3, the upper limit would of course decrease further significantly. 8 Thus, the upper limit shown above on Γ −1 (p → e + π 0 ), stemming from Eq. (13), is a robust and novel feature of the model. The predicted lifetime is accessible to proposed megaton size water Cherenkov (or equivalent) detectors.
Reversing the procedure given above, we can derive an upper limit on M eff and thereby on Γ −1 (p → νK + ). Owing to Eq. (13), this would correspond to the minimum allowed value of M X and r. Using central values of |α H | and α G (M X ) with p ≈ 4 (for concreteness), the observed lower limit on Γ −1 (p → e + π 0 ) ≥ 1.01×10 34 yrs [23] yields via Eq. (15): (M X ) min ≥ 6.26×10 15 GeV. This in turn 8 While Γ −1 (p → e + π 0 ) given by Eq. (14) does not explicitly depend on mq, mW , r and tan β, the upper limit on M X and thereby Γ −1 (p → e + π 0 ) does depend on these parameters via the correlation Eq. (13). The latter relates (M X ) max to (M eff ) min and thereby to the empirical lower limit on Γ −1 (p → νK + ) which depends on mq and mW . Because of this, the upper limit given in Eq. (21)  If we use central values of the parameters and the spectrum as noted above, with p ≈ 4 and tan β ≥ 3, the upper limit on Γ −1 (p → νK + ) (using Eq. (20)) would be 5.16 × 10 33 yrs. Allowing for uncertainties in the parameters in a combined manner, analogous to the case of d = 6 lifetime, we thus obtain 10 Γ −1 (p → νK + ) < ∼ (3.1 × 10 34 yrs) × In Eq. (22) the mild dependence of the curly bracket of Eq. (13) on mq and mW is not exhibited.
The actual lifetime is likely to be significantly lower than few ×10 34 yrs if M eff is not stretched to its upper limit (corresponding to e.g., M X > (M X ) min , or r > 1/300, or tan β > 3 and/or α 3 (m Z ) < 0.1196), or if mq < 1.5 TeV, or mW > 130 GeV. We thus find that not only the p → e + π 0 mode, but very likely even the p →νK + mode should be observable by improving the current experimental sensitivity by about a factor of five to ten.
Some important details concerning the present work, including those pertaining to the issues of fermion masses and mixings, and some variants as regards the cancelation of the U(1) A Fayet-Iliopoulos term, will be presented in a forthcoming longer paper [16].
In summary, we have presented a class of supersymmetric SO(10) models with low dimensional Higgs system that fully resolves all the naturalness issues of doublet-triplet splitting, including stability against higher order operators, generation of µ-term of order m susy , and proton stability.
The threshold corrections in these models are found to depend only on a few effective parameters, making the scenario very predictive. An intriguing feature of these models is the correlation equation and the corresponding constrained upper limit on Γ −1 (p → e + π 0 ). We find that in this class of models proton decays into both e + π 0 and very likelyνK + as well should show with an improvement in the current sensitivity by about a factor of five to ten. The building of a megaton water Cherenkov detector (or equivalent) would thus be most welcome.
We would like to thank Takaaki