Observable Neutron-Antineutron Oscillations in Seesaw Models of Neutrino Mass

We show that in a large class of supersymmetric models with spontaneously broken B-L symmetry, neutron--antineutron oscillations occur at an observable level even though the scale of B-L breaking is very high, v_{B-L} ~ 2 x 10^{16} GeV, as suggested by gauge coupling unification and neutrino masses. We illustrate this phenomenon in the context of a recently proposed class of seesaw models that solves the strong CP problem and the SUSY phase problem using parity symmetry. We obtain an upper limit on N-\bar{N} oscillation time in these models, \tau_{N-\bar{N}}<10^{9} -10^{10} sec. This suggests that a modest improvement in the current limit on \tau_{N-\bar{N}} of 0.86 x 10^8 sec will either lead to the discovery of N-\bar{N} oscillations, or will considerably restrict the allowed parameter space of an interesting class of neutrino mass models.


I. INTRODUCTION
It is widely believed that the most natural and appealing explanation of the recent neutrino oscillation results is provided by the seesaw mechanism [1] incorporated into extensions of the Standard Model that include a local B − L symmetry. The simplest models with local B − L symmetry are the left-right symmetric models [2] based on the gauge group SU(3) C × SU(2) L × SU(2) R × U(1) B−L . These models have the additional virtue that they explain the origin of parity violation in weak interactions as a consequence of spontaneous symmetry breaking in very much the same way as one explains the strength of the weak interaction in the Standard Model. Stability of the Higgs sector under radiative corrections calls for weak scale supersymmetry as in the Minimal Supersymmetric Standard Model (MSSM). It has recently been shown that if the MSSM is embedded into a left-right symmetric framework at a high scale v R ∼ 10 14 − 10 16 GeV, as suggested by neutrino oscillation data and by gauge coupling unification, it helps solve some important problems faced by the MSSM, viz., the SUSY CP problem [3], the strong CP problem [4] and the µ problem. Supersymmetric models with such a high scale embedding are therefore attractive candidates for physics beyond the Standard Model.
It was noted many years ago [5] that the electric charge formula of the left-right symmetric models, Q = I 3L +I 3R + B−L 2 , allows one to conclude from pure group theoretic arguments that parity symmetry breaking implies a breakdown of B − L symmetry as well with the constraint that 2∆I 3R = −∆(B − L). This simple relation is profoundly revealing. It says that the neutrinos must be Majorana particles since the lepton number breaking terms in the theory must obey |∆L| = 2 selection rule. This conclusion follows directly if Higgs triplets are used to break SU(2) R symmetry since I 3R = 1 for triplets, it also holds when Higgs doublets are used for this purpose, since gauge invariance requires the presence of two such doublets in the mass term for the neutrinos. Secondly, for purely hadronic baryon number violating processes, baryon number must change by at least two units, |∆B| = 2. This means that models based on left-right symmetric gauge structure can lead to the process where a neutron transforms itself into an antineutron (N −N oscillation [6,7,5]), while they may forbid the decay of the proton, which is a ∆B = 1 process.
While the above group theory argument predicts the existence of N −N oscillation in left-right symmetric models, its strength will depend on the details of the model. Using simple dimensional analysis it is easy to find that the lowest dimensional operators that contributes to N −N oscillation are six quark operators, a typical one being (u c d c u c d c d c d c ). This operator has dimension 9 and therefore the coupling strength scales as G ∆B=2 ∼ 1 M 5 , where M is the scale of new physics. It is natural to identify M with the scale of B − L (or parity) breaking. The current lower limit on N −N oscillation time, τ N −N ≥ 0.86 × 10 8 sec [8], 1 implies an upper limit G ∆B=2 ≤ 3 × 10 −28 GeV −5 . For N −N oscillations to be observable then, the scale M should be rather low, M ≤ 10 6 GeV.
One class of models where ∆B = 2 transition manifests itself through Higgs boson exchange has been discussed in Ref. [5]. There it was shown that if the SU(3) C × SU(2) L × SU(2) R × U(1) B−L model is embedded into the SU(4) C × SU(2) L × SU(2) R gauge group, then N −N oscillations can arise at an observable level if the SU(4) C breaking scale is in the 100 TeV range. In these models, N −N oscillation amplitude is intimately tied to an understanding of small neutrino masses via the seesaw mechanism as well as the breaking of quark-lepton degeneracy implied by SU(4) C symmetry. The same Higgs field that breaks SU(4) C and generates heavy Majorana masses for the right-handed neutrinos also mediate N −N oscillations here. With the scale of SU(4) C breaking in the 100 TeV range, these models would appear to be incompatible with gauge coupling unification. Furthermore, such a low scale of parity breaking would not yield naturally neutrino masses in the range suggested by current experiments. If we raise the scale of parity/SU(4) C breaking to values above 10 12 GeV, so that small neutrino masses in the right range are generated naturally, then N −N transition amplitude becomes unobservably small in these models. 2 Does the above arguments mean that N −N oscillations are beyond experimental reach based on current neutrino oscillation phenomenology? In this Letter we will show that this is not the case in a class of attractive seesaw models with local B − L symmetry. We will see that in these models a new class of ∆B = 1 operators is induced as a consequence of parity breaking. These operators lead to observable N −N oscillation despite the scale v R of parity breaking being close to the conventional GUT scale of 2 × 10 16 GeV. In fact, G ∆B=2 increases with v R and therefore one has the inverse phenomenon that increasing v R leads to stronger N −N oscillation amplitude. Interestingly, the scale v R implied by neutrino masses is such that N −N oscillation should be accessible experimentally with a modest improvement in the current limit. We obtain an upper limit of τ N −N ≤ 10 8 − 10 10 sec in this class of models. This prediction becomes sharper in a concrete model where flavor symmetries reduce considerably the uncertainties in the estimate of τ N −N . We emphasize that our upper limit is derived in the context of conventional seesaw models of neutrino mass without using any special ingredients to enhance N −N oscillation amplitude. This should provide new impetus for an improved experimental search for N −N oscillations.

II. HIGH SCALE SEESAW MODEL AND N −N OSCILLATION
The basic framework of our model involves the embedding of the MSSM into a minimal SUSY left-right gauge structure at a scale v R close to the GUT scale. The electroweak gauge group of the model, as already mentioned, is SU(2) L × SU(2) R × U(1) B−L with the standard assignment of quarks and leptons -left-handed quarks and leptons (Q, L) transform as doublets of SU(2) L , while the right-handed conjugate ones (Q c , L c ) are doublets of SU(2) R . The 2 A counter example where a higher scale of parity violation can go hand in hand with observable N −N oscillation was noted in the context of a SUSY SU (2) L × SU (2) R × SU (4) c model in Ref. [10]. These models possess accidental symmetries that lead to light (∼ 100 GeV) diquark Higgs bosons even though the scale of parity violation is high. As a result, the N −N oscillation operator can have observable strength. Unification of gauge couplings is however difficult to achieve in these models.
It has recently been shown that if there exists a Z 4 R symmetry, the minimal model just described will solve the strong CP problem and the SUSY phase problem based on parity symmetry [4]. Furthermore, the µ term will have a natural origin. One possible Z 4 assignment was given in Ref. [4]. Here we present a slight variant, which yields the same superpotential at the renormalizable level as in Ref. [4] and thus preserves all its success. Under this Z 4 , the superpotential W changes sign, as do d 2 θ and d 2θ . The quark fields The gauge invariant superpotential consistent with this Z 4 R symmetry at the renormalizable level is This superpotential breaks the gauge symmetry to that of the Standard Model in the SUSY limit without leaving any unwanted Goldstone bosons and induces realistic quark masses and mxings. The baryon number violating processes as well as neutrino masses arise in this model from higher dimensional operators induced by Planck scale physics. They will be the main focus of the rest of the paper. We shall pay special attention to the relation between the neutrino mass and the N −N oscillation time. The relevant dimension four operators in the superpotential which are scaled by M −1 Pl and are allowed by the Z 4 symmetry are: Operator O 1 gives rise to Majorana masses for ν R of order v 2 R /M Pl . Combining this with the Dirac neutrino masses arising from Eq. (1), small neutrino masses will be generated by the seesaw mechanism. For v R ∼ 10 14 − 10 16 GeV, the magnitude of the light neutrino masses are in the right range to explain the atmospheric and the solar neutrino oscillation data. Operator O 2 , which is also invariant under the Z 4 , leads to baryon number violation. While O 1,2 could have their origin in quantum gravity, they may also be induced by integrating out vector states that have Z 4 -inviariant masses of order the Planck scale.
Note that operators such as LΦχ c , LLL c χ c , QLQ c χ c are not allowed by the Z 4 symmetry. If they were present along with O 2 , they would lead to rapid proton decay. Note also that the well known proton decay operator QQQL is not allowed by the Z 4 symmetry. In any case its presence would not have been a problem since it is scaled by the Planck mass and therefore can lead to a proton lifetime consistent with the present lower limit.
To see the connection between neutrino masses and the N −N oscillation time τ N −N ≡ (1/δm N −N ) qualitatively, first we note that the operator O 1 leads to the Majorana mass for the right handed neutrino M R = where C is a dimensional constant which depends only on the details of weak scale physics and does not involve the high scale v R . We will evaluate C in the next section. This simple relation makes it clear that our present knowledge of the neutrino masses allows a direct prediction of the N −N oscillation time in the context of the supersymmetric left-right models broken by doublet Higgs fields.

III. FROM SUPERSYMMETRIC ∆B = 1 OPERATOR TO N −N OSCILLATIONS
Let us now proceed to examine the expected N −N oscillation time resulting from the ∆B = 1 operator O 2 of Eq. (2). An important point to note here is that since O 2 is a superpotential term with antisymmetric color contraction it must have antisymmetric flavor contraction as well. The flavor structure of this operator is then of the type u c d c s c , u c d c b c or u c s c b c in terms of the superfields. We must then use flavor mixings to obtain the fermionic operator of the type u c d c d c and then the six quark N −N operator. The dominant contribution to this process comes from the Feynman diagram shown in Fig. 1 which proceeds through the exchange of a gluino and squarks [11] and involves twod c −b c mixings. The strength of the ∆B = 2 operator resulting from Fig. 1 can be estimated to be where Mg is the gluino mass, mq is the squark mass and (δ 13 RR ) is thed c −b c mixing angle. The effective baryon number violating u c d c b c Yukawa coupling in the superpotential is parametrized here as f ′ (v R /M Pl ) (see Eq. (2)).
Let us first discuss the origin of the flavor mixing that changes a u c d c b c operator to the required u c d c d c operator. The dominant source for this in the present context turns out to be the mixing ofb c withd c . Such mixings occur in the left-right supersymmetric model since the right-handed quark mixings are physical above the scale v R . The renormalization group evolution of the soft SUSY breaking mass parameters between M Pl and v R will then induce mixings in the right-handed down squark sector proportional to the top-quark Yukawa coupling and the right-handed CKM mixings. This is analogous to the RGE evolution in the MSSM inducing squark mixing in the left-handed down squark sector proportional to the left-handed CKM angles and the top-quark Yukawa coupling. We estimate this right-handedd c −b c mixing to be This estimate is obtained by integrating out the RGE between M Pl and v R assuming universality of masses at M Pl [12]. Since above v R , both t c and b c are part of the same SU(2) R multiplet, unlike in the MSSM, b c Yukawa coupling is of order one. In this momentum range, the top-quark Yukawa coupling reduces the mass ofb c . In going to the physical basis of the quarks, this effect will induce the squark mixing quoted in Eq. (5). For the numerical estimate we took m 0 = M 1/2 and A 0 = 0 and v R ∼ 2 × 10 16 GeV for illustration. There is a second source of flavor violation that inducesd c −b c mixing in general SUSY models. That is the baryon number violating Yukawa couplings themselves. If we write in standard notation, the effective B-violating superpotential arising from Eq. (2) Recalling that λ ′′ ∼ v R /M Pl , we see that while this source of flavor mixing may not be negligible, it would be typically smaller than the ones from the right-handed quark mixings of Eq. (5). A third source of flavor violation relevant for N −N oscillations has been identified in Ref. [14] involving the exchange of the Wino. Such diagrams will have an electroweak loop suppression and a chirality suppression necessary to convert the left-handed squark to the right-handed one. We find that this contribution to δm N −N has a suppression factor given approximately by [(α 2 /4π)(m b /mq)] 2 ∼ 1 × 10 −9 (valid for small tan β) which is about two orders of magnitude smaller in this class of models compared to the gluino exchange diagram of Fig. 1.
One has to calculate the hadronic matrix element of the six quark operator in order to obtain the τ N −N . This has been discussed in several places in the literature [15]. The calculations of this "conversion" factor can be done using crude physical arguments, according to which one has to multiply the G ∆B=2 by |ψ(0)| 4 to obtain δm N −N where ψ is the baryonic wave function for three quarks inside a nucleon. On dimensional grounds, one can deduce that |ψ(0)| 4 ≃ Λ 6 QCD , which implies that δm N −N ∼ 10 −5 G ∆B=2 GeV. More detailed bag model calculations have been carried out. Rao and Shrock in Ref. [15] quote this conversion factor to be 2.5 × 10 −5 G ∆B=2 . We shall use this number for our numerical illustrations.
Combining this matrix element with Eq. (4)-(5) we obtain We can rewrite Eq. (7) in a form that makes the connection with the neutrino mass more transparent. The mass of ν τ can be expressed through the seesaw formula from Eq.
Since the value of m ντ can be determined from the atmospheric neutrino data under certain assumptions, we conclude that within the seesaw framework, measurement of N −N oscillation will be a measure of the Dirac mass of the tau neutrino. This can then be used as a way to discriminate between models of neutrino masses.
To see the specific prediction for N −N oscillations within the context of the class of models under consideration, we need to know the ν τ Dirac mass. We can estimate it from the following relations for the Dirac masses of the third generation quarks and leptons in the SUSY left-right model: Here α u,d are the Higgs mixing parameters obtained from Eq. (1) (Eg: tan α u = λ 1 /λ 2 ) and v u,d are the VEVs of the MSSM doublets. From Eq. (9) it follows that in the limit of h t,1 ≫ h t,2 and h τ,1 ≫ h τ,2 , we get m ν D τ ≃ m τ mt m b . Since at such high scales m b ≃ m τ , this predicts m ν D τ ≃ m t . In fact, we find that unless the two terms in Eq. (9) for m ν D τ are precisely canceled, the Dirac mass of ν τ will be approximately equal to m t .
Using m ν D τ ≃ m t and f ∼ 1, f ′ ∼ 10 −1 , we get a value for the N −N oscillation time which is tantalizingly close to the present experimental lower limit [8]. For values of m ν D τ 10 times smaller than m t , and taking the supersymmetric particle masses as large as 1 TeV, we see that τ N −N is less than 9 × 10 9 seconds, 3 which is in the range accessible to a recently proposed experiment [17]. It would thus appear that a search for neutronantineutron oscillation will provide an enormously useful window into neutrino mass models and as such a powerful constraint on the nature of new physics beyond the Standard Model.
The prediction for N −N oscillations can be sharpened if we make use of flavor symmetries to determine the coefficients f and f ′ in Eq. (8). We illustrate this with a specific choice of flavor symmetry [18] taken to be SU(2) H × U(1) H . The first two families of fermions form doublets of SU(2) H and have a U(1) H charge of +1 while the third family fermions are singlets under both groups. This flavor symmetry is broken by a pair of doublets φ(1)+φ(−1) and singlets χ(1) +χ(−1). Allowing for effective operators suppressed by a scale M larger than the VEVs of these fields provides a natural explanation of the fermion mass and mixing angle hierarchy. If we choose φ = φ = ǫ φ M and χ = χ = ǫ χ M, a resonable fit to all quark and lepton masses is obtained, including neutrinos, for ǫ φ ≃ 1/7 and ǫ χ ≃ 1/20 and all dimensionless couplings being order one [18]. In this model, we can estimate the couplings f and f ′ from the horizontal quantum numbers. They are f ∼ ǫ 2 φ and f ′ ≃ ǫ 2 χ sin θ C , so that 4 f /f ′2 ≃ 8 × 10 4 This estimate leads to τ N −N ≃ 2 × 10 9 sec from Eq. (8). Allowing for uncertainties of order 1 in this estimate, we expect that τ N −N not to exceed about 10 10 sec.
Before we conclude a few comments are in order: 1. The model becomes unacceptable as soon as SU(3) C × U(1) B−L is embedded into a higher symmetry such as SU(4) C or SO(10) group because in that case, the ∆B = 1 operator described in Eq. (2) is accompanied by other R-parity violating operators coming from the same higher dimensional operator O 1,2 due to the higher symmetry. Together, they would lead to unacceptable proton decay rate. Thus observation of N −N oscillation would be a signal of an explicit SU(3) C × U(1) B−L symmetry all the way upto the Planck (or string) scale.
2. Baryogenesis has to proceed through a weak scale scenario since the ∆B = 1 interactions in the model are in equilibrium down to the TeV scale and will wash out any primordial baryon or lepton asymmetry. We note that the baryon number violating interactions contained in O 2 themselves can potentially be the source of weak scale baryogenesis [19].
3. The lightest neutralino in this model is unstable and will decay via χ 0 → qqq modes due to the presence of the effective ∆B = 1 operator O 2 . This prediction is directly testable at colliders. An alternative candidate for dark matter must be sought.

IV. CONCLUSIONS
In conclusion, we have found that in a large class of seesaw models for neutrino masses, despite the high scale of sessaw dictated by the current neutrino oscillation data, neutronantineutron oscillation is in the observable range. In fact, unless the Dirac masses of neutrinos are far below those deduced under simple and reasonable assumptions, we predict an upper bound on the neutron-antineutron oscillation time in the range of 10 9 − 10 10 sec. This is very close to the present experimental lower limit on N −N oscillations. In the most conservative theoretical scenario, the measurement of N −N oscillation time would be a measure of the Dirac mass for the tau neutrino, given the values of squark masses. This in itself would be an extremely interesting result, since it would discriminate among theoretical models of neutrino masses. This is apart from the fundamental importance that any observation of baryon number violation will carry. We therefore strongly urge a new experimental search for neutron-antineutron oscillation.