Double beta decay of (cid:1) − hyperons

We compute the strangeness-conserving double beta decay of (cid:1) − hyperons, which is the only hadronic system that can undergo such decays. We consider both, the lepton number conserving (cid:1) − → (cid:1) + e − e − ¯ ν ¯ ν (ββ (cid:1) 2 ν ) and the lepton number violating (cid:1) − → (cid:1) + e − e − (ββ (cid:1) 0 ν ) modes. The branching ratios of these ββ decays are suppressed at the level of 10 − 30 considering a light neutrino scenario in the case of the ββ (cid:1) 0 ν channel. The dynamical origin of such low rates and their possible enhancements are brieﬂy discussed. Given its simplicity those decays can be used also for the purposes of illustrating the main features of double beta decays. 


Introduction
Neutrinoless double beta (ββ 0ν ) decays would occur if a mechanism allows the violation of the total lepton number L by two units. Their observation in experiments will provide unambiguous evidence for physics beyond the Standard Model (SM). At present, the first evidences for physics beyond the SM come from the flavor oscillations in the neutrino sector that required to explain the deficit observed in solar [1] and atmospheric neutrinos [2]. Flavor oscillations of neutrinos do not require a change of the total lepton number (namely, | L| = 0), allow us to conclude that neutrinos are massive, but do not establish whether they are Dirac or Majorana particles. If neutrinos are Majorana particles, ββ 0ν decays have to occur at some level; their observation will establish the Majorana nature of neutrinos beyond any doubt.
The strangeness-conserving ( S = 0) ββ 0ν decays have been unsuccessfully searched in nuclear transitions for several decades. An intense activity in the experimental and theoretical fronts [3][4][5][6][7][8] witness the importance of such decays as a sensitive probe of physics beyond the SM. Some examples of extensions of the SM that can induce contributions to ββ 0ν decays are right-handed weak couplings as well as the Higgs exchange [9], right-handed weak coupling involving heavy Majorana neutrinos [10], massless Majoron emission [4,[11][12][13][14], and R-parity breaking in the supersymmetric models [15,16]. From a theoretical point of view, ββ 0ν , decays in nuclei are limited in precision due a wide range of model-dependent predictions for the nuclear wavefunctions. In the present Letter, we study the double beta (both the lepton number-conserving ββ 2ν and lepton number-violating ββ 0ν ) strangenessconserving decays of the − hyperon. The − hyperon is a unique system in hadron physics that can undergo strangeness-conserving double beta decays as it will be explained below. The hadronic matrix elements necessary for such calculations are well known and, therefore, can exhibit the underlying mechanisms for double beta decays in a more clean way.
As we have mentioned before, the determination of an upper bound for the effective neutrino mass in nuclear ββ 0ν decays is limited by model-dependent evaluations of the nuclear matrix elements. Some of the difficulties we encounter in those calculations are the following: 1. The nucleus is a many body system with many degrees of freedom; in practice there is not a well defined rule to choose the most relevant components to describe an specific excitation; 2. The Hilbert space where nuclear models are worked out have a huge dimension requiring a lot of time consuming computational work; and 3. The multipole expansion for the ββ 0ν decay amplitude is rather complex making theoretical expressions difficult to manipulate.
Despite these limitations in theoretical inputs, the large sensitivity of present experiments have been able to set strong constrains on the so-called effective Majorana mass term, m ee ≡ U 2 el m ν l , where m ν l denote neutrino mass eigenstates. By assuming that ββ 0ν in nuclei are mediated by the exchange of light Majorana neutrinos, the experimental upper bound on m ee is 0.2 eV [17].
Before we proceed with our calculation, it is interesting to take a look at other reactions that can provide information on the violation of lepton number. The upper limits available on these rare processes can be used to set upper limits on the matrix elements m αβ ≡ U αl U βl m ν l , where α, β = e, µ, τ . Thus, muon to positron conversion in nuclei µ − + (A, Z) → (A, Z − 2)e + [4,18] gives m eµ < 17(82) GeV depending on the spin of the initial proton pair; the production of three muons in neutrino-nucleon scattering [19] leads to the upper limit m µµ 10 4 GeV; this limit has been slightly improved at HERA through the reaction e + p →νl + 1 l + 2 X [20], giving m µµ 4 × 10 3 GeV and also for first time limits on the m lτ (connected with the τ -sector) were given; finally, the non-observation of heavy Majorana neutrinos at various colliders [21] can also be used to set limits on the effective Majorana mass. Similarly, some rare kaon decays are also useful to constrain lepton number violating interactions. For instance, present bounds on the branching ratio of the K + → µ + µ + π − decay [22] translates into the upper limit m µµ 4 × 10 4 MeV [23]. Those bounds from collision experiments and rare kaon decays are several orders of magnitude above the limit m µµ 4.4 eV, inferred by an analysis [24] that combines experimental constrains from atmospheric and solar neutrino oscillation and the tritium beta decay endpoint experiment [25,26].
Conversely, we can use the existing bounds from lepton number violating processes to set upper limits on the branching ratios of some rare kaon decay. Thus, the upper limit B(K + → µ + µ + π − ) 10 −30 (10 −19 ) [27] can be obtained in models with a light (heavy) neutrino scenario, while B(K + → e + e + π − ) 10 −26 [28] can be derived from upper bounds from nuclear ββ 0ν decays. These values are well below the sensitivities of current experiments; for example the limits reported by the E865 experiment [23] are B(K + → e + e + π − ) < 6.4 × 10 −10 , and B(K + → µ + µ + π − ) 3 × 10 −9 . This shows that we are far from detecting such processes, in spite of the special window in the hundred MeV region [27], and that the sensitivity of such processes are well below the previously discussed ββ 0ν case. Nevertheless, it is important to pursue searches for | L| = 2 processes since they would lead to nonvanishing results even if nuclear ββ 0ν decay turns out to vanish or becomes extremely suppressed. In this Letter we study the two double beta decays of the − hyperon, namely: the lepton number conserving − → + e − e −νν (ββ 2ν ) and the lepton number violating − → + e − e − (ββ 0ν , L = 2) channels. The Feynman graphs for the ββ 2ν and ββ 0ν decays, indicating the two intermediate states, are shown in Fig. 1(a) and (b), respectively. To our knowledge, these decay modes have not been reported previously in the literature and even an experimental upper limit is not available. The isotriplet of hyperons ( + , 0 , − ) is a unique system of hadrons that can undergo strangeness-conserving double beta decays. Actually, the − and the + are not antiparticles of each other; they have a mass splitting of m − − m + = 8.08 MeV [29], which allows a sufficient phase space for ββ decays. An interesting feature of this system is that we can identify only two well defined intermediate states ( 0 and ) that can give important contributions to the ββ transitions under consideration. One of these states (the 0 ) lies in the middle of the − , + levels while the other (the ) is around 77 MeV below them. Namely, one of these intermediate states can be real while the other is virtual.
On another hand, the hadronic matrix elements required for the calculations are dominated by a couple of axial and vector form factors, which have been computed by several authors in the literature [30,31] with good agreement among them. This is an important advantage over the evaluation of nuclear transition matrix elements. All these characteristics of the hyperon system make very interesting the study of their double beta decays, even as a possible textbook example to learn the formalism of ββ transitions applied to free baryons. In the following we consider each case separately.

Lepton number-conserving ββ decay
One of the four Feynman graphs corresponding to the ββ 2ν decay is shown in Fig. 1(a) (the other three contributions are obtained under proper antisymmetrization with respect to final state electrons and antineutrinos).
The effective four-fermion weak Hamiltonian acting at each vertex has the usual current-current form [8,31] (1) is the baryonic current operator underlying the B → B transition and is the V − A leptonic current operator. 1 In Eq. (2), f 1 and g 1 are dimensionless vector and axial-vector form factors for the B → B transition. In our approximation we have neglected their momentum transfer dependence and we Table 1 Hyperon form factors at zero momentum transfer (see Table III in Ref. [31]) have also neglected the small contributions of the induced magnetic and scalar form factors for the vector and axial currents. The effective weak coupling constant is is the Fermi constant and V ud the relevant element of the Cabibbo-Kobayashi-Maskawa mixing matrix. The rate for the ββ 2ν decay from the initial − ≡ A to the final + ≡ B hyperon is given in the rest frame of A by (we use natural units, i.e.,h = c = m e = 1) where m A ( B ) is the energy of the initial (final) hyperon, and i and p i (ω i and q i ) denote the energy and momentum of the electron (neutrino). The decay amplitude reads is the baryonic matrix element between states with momentum p I and p F , and is the leptonic tensor. The values for the form factors f A η ≡ f 1 ( − η), g A η ≡ g 1 ( − η), f B η ≡ f 1 (η + ) and g B η ≡ g 1 (η + ) at zero momentum transfer are summarized in Table 1.
Given the small mass difference between the relevant hyperon states, we can use the non-relativistic impulse approximation for the baryonic current [8,32]. The small mass difference between the hyperon states is responsible of the suppression of the decay via the real intermediate 0 particle, while the decay through an intermediate on-shell state is forbidden since m < m + . Keeping only the usually called Fermi and Gamov-Teller operators we have where s I and s F denote the spin of the initial and final baryons in the I → F transition. In the spirit of the nonrelativistic approximation the energy denominator in Eq. . This singularity can be cured by taking into account the finite width (Γ 0 = 8.89 keV [29]) of the 0 intermediate state. Therefore, we will define: • In the case η = 0 : • In the case η = : After a lengthy but straightforward calculation we can write the decay rate as follows: The C j (ηη ) terms denote quartic combinations of form factors, and I j (ηη ) are the phase space factors defined as follows: The numerical values for the factors entering in the expression of the decay rate, Eq. (12), are given in Table 2. We can check that the main contribution comes from the term C 1 ( 0 0 )I 1 ( 0 0 ), which includes the contribution of a real 0 hyperon intermediate state.
Using the values obtained in Table 2, we can compute the branching ratio from the rate in Eq. (12). We obtain: Just for comparison we have shown within parenthesis the value corresponding to the contribution of the 0 intermediate state. As expected, this contribution dominates almost completely the decay rate. The branching ratio given above turns out to be very suppressed due essentially to the large decay width of the 0 hyperon appearing in Eq. (10). As we know, the decay rate of the 0 is 10 orders of magnitude larger than those of the charged hyperons because it can undergo the electromagnetic decay 0 → γ . Table 2 Numerical values for C j (ηη ) and I j (ηη ) (in units of MeV 9 ) for ββ 2ν decay

Neutrinoless double beta decay
The decay rate for the (three-body) neutrinoless ββ 0ν mode reads The decay amplitude corresponding to the diagram in Fig. 1(b) (after proper anti-symmetrization with respect to identical electrons) is The four-momentum of the intermediate state η is Q(q) ≡ p A − p 1 − q = p B + p 2 − q and its energy is (17), and expanding the weak neutrino eigenstate as a mixture of light massive Majorana neutrino states, i.e., uν(q) = l U el u ν l (q), we get with the leptonic factor (19) L µν (p 1 , p 2 ) =ū e (p 1 )γ µ (1 − γ 5 )γ ν u C e (p 2 ). The factor I η (p 1 ) in Eq. (18) corresponds to the following loop integral (Q(q) = p A − p 1 − q): which has a logarithmic divergence that we will manipulate in a simple cutoff procedure. After using Feynman parametrization techniques [33], we get where Λ c is the cutoff energy, and we have defined We stress here that the origin of this logarithmic divergence is related to the effective vertices we are using for the hadronic form factors. This divergence can in principle be cured by including the weak form factors which are expected to fall as 1/[(p A − Q(q)] 2 with q 2 → ∞ in the dipole approximation, but their real behavior in the large q 2 limit are actually determined by QCD. In the case of the nuclear ββ 0ν decays it is usual to assume that the divergence is driven by the internucleon distance, which sets Λ c ∼ 1 GeV. This value can be identified with the largest momenta that the neutrino can carry which in turn is fixed by the lowest distance between two nucleons in the nucleus (Λ c ∼ (2d) −1 , where d is the nucleon radius). We will assume here that the average distance between the quarks within the hyperons are of the order of a typical hyperon (or nucleon) radius. Here, we adopt the point of view that our results are stable as far as they do not depend strongly on the specific value of the cutoff Λ c .
The integration in Eq. (21) can be simplified by neglecting in Eq. (22) all lepton masses and momenta, which is consistent for the neutrino since we are assuming a light neutrino scenario. We obtain In the above expressions, we have introduced the following dimensionless constants: In order to evaluate the unpolarized squared amplitude we will use the non-relativistic impulse approximation for the final baryon and take p B (m B , 0) (we cannot do the same approximation for the intermediate baryon state in this case). The decay rate for the ββ 0ν transition becomes: where m ee = l m ν l U 2 el is the effective neutrino mass. The other factors appearing in Eq. (25) are defined as follows: for the product of form factors, and for the phase space integral.
The neutrinoless double beta decay rate depends on the cutoff Λ c and the neutrino effective mass m ee , which are free parameters in our model. Based on present bounds on electron neutrino mass ∼ eV [17], we show in Fig. 2 the decay rate Γ 0ν as a function of the cutoff Λ c , for a fixed neutrino mass of 10 eV. From this figure we can clearly see that our results are not very sensitive to the specific cutoff value in the region under consideration, which gives further support to the method employed for the logarithmic divergence.
For illustrative purposes let us use Λ c = 1 GeV, which corresponds to a rough estimate of the inverse of the size of hyperons. The branching ratio in this case becomes Thus, the neutrinoless double beta decay of the − hyperon is very suppressed and even smaller that the branching ratios of ββ 0ν decays of kaons. Note, however, that in the case of hyperons we do not have a single neutrino (tree-level) intermediate state contribution as in the case of neutrinoless double beta kaon decays [27].

Summary and conclusions
We have considered the strangeness-conserving double beta decays of − hyperons. Several characteristics make this particle a unique and interesting system to study (and to learn) such decays as an alternative to the corresponding decays in nuclei. First, the isotriplet of hyperons is the only system of hadrons that can undergo double beta decays since the I 3 = +1, −1 components have different masses. Second, we can identify a few intermediate states that dominate such decays where one state is in the middle and the other is below the initial and final hadronic levels. Finally, the uncertainties in hadronic matrix elements are much smaller than in the nuclear case.
We study both the lepton number-conserving ββ 2ν and the lepton number-violating ββ 0ν decays. For the lepton number conserving decay we obtain B(ββ 2ν ) = 1.38 × 10 −30 and for the neutrinoless decay we get B(ββ 0ν ) = 1.49 × 10 −35 , standing in the light neutrino scenario in the last case. The suppression of the lepton number-conserving decay is due to the large decay width of the 0 intermediate state. Therefore, it would be nice to find an analogue system to the hyperons (the b baryons?) where the mechanism under consideration could produce an enhancement of the decay rate if the corresponding isotriplet is similar to the spectra of hyperons.
Our result for the neutrinoless double beta decay is almost insensitive to the specific value of the cutoff parameter used to regulate the divergent integrals.
Our numerical results for the branching ratios may look discouraging. However, let us assume an hypothetical model where B(ββ 0ν ) = (10 −20 ∼ 10 −25 ) m ee 2 ; in this very optimistic scenario, an experimental upper limit of 10 −8 would translate into the interesting bound m ee 1 ∼ 300 MeV. On another hand, if there exists such a model that causes − to have faster neutrinoless double beta decays, the SM background due to the lepton number-conserving double beta decays would certainly be very small.