Light neutrino and heavy particle exchange in $\nbb$-decay

A simple and precise method is presented to compare contributions to neutrinoless double-beta decay ($\nbb$-decay) from heavy particle exchange and light Majorana neutrino exchange. This procedure makes no assumptions about the momentum transfer between the two nucleons involved in the $\nbb$-decay process. It is shown that for a general particle physics model, the characteristic $\nbb$-decay scale $>4.4$ TeV when all the coupling constants are assumed to be natural and of ${\cal{O}}(1)$.

The observation of 0νββ-decay would shed light on the neutrino mass magnitude and whether it is Dirac or Majorana, but additional input would be required. Indeed, a number of particle physics model beyond the standard model (SM) have lepton-number violating (LNV) operators that allow 0νββ-decay through the exchange of heavy particles such as the neutralino [4,5,6] or a heavy right-handed neutrino [7,8,9]. Thus, the observation of 0νββ-decay would provide a unique window on physics beyond the SM with broad implications for LNV particle physics models, and the way neutrino masses are generated within them. It follows that disentangling contributions to 0νββ-decay due to light Majorana neutrino exchange from the heavy particle contributions is crucial if one is to use this data to constrain the neutrino mass matrix and the models that generate it and LNV.
Although it has been known for some time that heavy particle exchange and light Majorana neutrino exchange contributions to 0νββ-decay can be comparable, the comparisons have usually been performed by making assumptions about the momemtum flow through the light Majorana neutrino and estimating orders of magnitude [10,11]. In this current work, a simple and more precise procedure is presented to compare the relative importance of both processes to 0νββ-decay. Operators stemming from light neutrino exchange that have precisely the same form as the leading order (LO) heavy particle exchange 0νββ-decay operators [12] are derived and used for the comparison. Hence, there is no need to make an estimate of the average momentum flowing through ν M in Fig. 1a. This might seem counter-intuitive since 0νββ-decay mediated by light neutrino exchange is suppressed by the ratio of the neutrino mass to Q 2 (the momentum squared flowing through the neutrino propagator |Q| ∼ 100 MeV). 1 In contrast, 0νββ-decay is suppressed by Λ ββ when it occurs through heavy particle exchange, where Λ ββ is the heavy scale (typically of the order of 1 TeV or larger) that characterizes the strength of the 0νββ-decay operator.
The SM low-energy effective Lagrangian with Majorana neutrinos gives rise to the 0νββ-decay operator of Fig. 1a where a light Majorana neutrino is exchanged. In Eq. (1), is the Fermi constant, M W the charged weak boson mass, g = 0.65265, u and d are the up and down quark fields respectively, e the electron field and ν the neutrino field. From the Feynman rules, the amplitude for this diagram is simply where m ν is the neutrino mass. The current-current interaction of Eq. (1) gives rise to lepton-hadron vertices (πνe, N N νe) that contribute to 0νββ-decay through the operators shown in Fig. 2 Operators that contribute to 0νββ-decay are either suppressed or enhanced by powers of (p/Λ H ) n (where n is the chiral power of the operator) with p ∼0.1 GeV and where Λ H is a hadronic scale ∼1 GeV. The chiral power of a 0νββ-decay operator can be calculated with the following rules: • a derivative in a vertex counts as one power of p; • pion and light neutrino propagators count as p −2 .
Considering that the parity-conserving pion-nucleon vertex is (g A /f π )N γ µ γ 5 N ∂ µ π and noting the derivative in the pion-lepton operator in Eq. (3), one finds that the 0νββ-decay operators of Fig. 2 are all of O(p −2 ). From Ref. [12], it is seen that these operators have the same chiral power as the LO 0νββ-decay heavy particle exchange operators. This observation suggests a more precise method to compare heavy and light particle exchange contributions to 0νββ-decay.
Consider the amplitude for the Feynman graph of Fig. 2a: where the error stemming from writing MeV being the energy carried off by the electrons. The approximation in Eq. (4) is therefore very good. Eq. (4) has the same form as the LO 0νββ-decay hadronic operators stemming from the exchange of a heavy particle [12]. It follows that to a high degree of precision, one can introduce a new "short-distance" 0νββ-decay operator that stems from light neutrino exchange: This operator combined with the parity-conserving pionnucleon vertex yields Eq. (4). In this form, comparing heavy and light particle exchange contributions to 0νββdecay is relatively easy. The only caveat is the existence of a possible suppression of Fig. 2a relatively to Fig. 2c due to the fact that the pion has a finite range smaller than the size of the nucleus, while the neutrino exchanged in the latter graph does not. In coordinate space, the suppression of the nuclear matrix elements will occur through exponentials of the form e −mπr . In momentum space, the suppression is due to a factors of the form Q 2 /m 2 π . The way to handle this is discussed further below. From Ref. [12], a LO operator has the form where the general 0νββ-decay vertex is assumed to be suppressed by five powers of the 0νββ-decay scale, Λ ββ as occurs in many popular particle physics models. 2 For 2 Note that the authors of Ref [11] insert an extra factor of G 2 F M 4 W in Eq. (6). example, this suppression by five powers occurs in Rparity violating supersymmetry (RPV SUSY) and the left-right symmetric model (LRSM). 3 This LO operator leads to the ππee operator where β is a parameter of O(1) generated by the hadronization of the quark currents as discussed in Ref. [12]. Comparing Eq. (7) with Eq. (5), it is seen that the contributions to 0νββ-decay from light neutrino exchange and heavy particle exchange processes are equal when where α M is a number that takes into account the fact that the matrix elements of the pion exchange diagram of Fig. 2a is suppressed with respect to the matrix element of the operator generated by the graph of Fig. 2c. In Ref. [14], the nuclear matrix elements from pseudoscalar couplings (corresponding to Fig. 2a) and axialvector coulings (corresponding to Fig. 2c) were computed for nine nuclei and tabulated in their Table II. On average, for light neutrino exchange, the matrix elements stemming from pseudoscalar coupling and denoted M light P P in Ref. [14] are ten times smaller than the matrix elements stemming from axialvector coupling and denoted M light AA . The value α M = 10 will therefore be used. Limits on Λ ββ derived below are not very sensitive to the exact value of α M since Λ ββ appears to the fifth power in Eq. (8): factors of two or three in α M can change the results appearing below by at most 1.2.
Eq. (8) is plotted in Fig. 3. Above the LHE line, light neutrino exchange is larger than the heavy particle contribution to 0νββ-decay; the reverse is true below the LHE line.
The dashed line in Fig. 3 is the upper-limit on m ν < 0.23 eV from the WMAP [15]. The upper-limit on m ν implies that the 0νββ-decay heavy particle exchange operator is larger than the light neutrino exchange contribution for indicated by the first arrow in Fig. 3. This is essentially a model independent limit. If one uses the neutrino mass 3 In RPV SUSY, the coupling has the form λ ′2 111 /Λ 5 S (λ ′ 111 is the RPV SUSY coupling constant and Λ S is the SUSY breaking scale) while in the LRSM it has the form ζg 4 /(32M 2 W M 2 R N R ) where M R is the mass of the right-handed boson, N R is the mass of the right-handed neutrino, and ζ < M 2 W /M 2 R is the mixing angle between W R and W L . limit m ν < 0.04 eV that could be reached by the future Planck mission [16], and represented by the dotted line in Fig. 3, then as indicated by the second arrow in the plot. We can evaluate in specific models the point at which heavy particle exchange becomes larger than light neutrino exchange. Considering the LRSM first, we have [8,9,12] where the limits ζ < 10 −3 , M R > 800 GeV on the weak gauge boson mixing angle and the right-handed weak boson mass were used in evaluating Eq. (12). Thus, the upper-limit on the right-handed particle masses below which the heavy particle exchange contribution dominates is M R ∼ N R ∼ Λ R < 0.9 TeV. Using the lower-limit on the half-life of 0νββ-decay of T 0νββ 1/2 > 10 25 years, one obtains a lower-bound on the mass of the heavy righthanded particles of ∼ 1 TeV [12]; this implies that the region in the LRSM parameter space where the heavy particles dominate over the light neutrino exchange contributions is essentially ruled out for the current limits on m ν .
Using the limit in Eq. (10) instead, one obtains instead.
Note that with current limits on 0νββ-decay, one requires the coupling constant λ in Eq. (6) to be ≪ 1 if one demands that Λ ββ ∼ = 1 TeV. This observation is here verified in the LRSM and RPV SUSY. Although such a small value of λ is clearly allowed, it should be explained since naturalness suggests λ ∼ 1 instead. In this case, one would expect Λ ββ > 4.4 TeV.
In this note, a simple and precise method of comparing contributions to 0νββ-decay was presented. It was shown that LO ππe − e − operators can be written down for both light neutrino exchange and heavy particle exchange contributions to 0νββdecay. This observation facilitated their comparison and allowed us to plot a graph in the neutrino mass and heavy particle scale Λ ββ parameter space to discern the regions where one contribution may be larger than the other. Using current limits on 0νββ-decay, it was also shown that Λ ββ 4.4 TeV in a general particle physics model where the 0νββ-decay operator coupling constant is assumed to be of O(1).
The author would like to thank Petr Vogel and Vincenzo Cirigliano for useful discussions and comments in the preparation of this manuscript.