Left-right symmetry and leading contributions to neutrinoless double beta decay

We study the impact of the mixing (LR mixing) between the standard model $W$ boson and its hypothetical, heavier right-handed parter $W_R$ on the neutrinoless double beta decay ($0\nu\beta\beta$-decay) rate. Our study is done in the minimal left-right symmetric model assuming type-II dominance scenario with charge conjugation as the left-right symmetry. We then show that the $0\nu\beta\beta$-decay rate may be dominated by the contribution proportional to this LR mixing, which at the hadronic level induces the leading-order contribution to the interaction between two pions and two charged leptons. The resulting long-range pion exchange contribution can significantly enhance the decay rate compared to previously considered short-range contributions. Finally, we find that even if future cosmological experiments rule out the inverted hierarchy for neutrino masses, there are still good prospects for a positive signal in the next generation of $0\nu\beta\beta$-decay experiments.

Determining the properties of the light neutrinos under charge conjugation is a key challenge for particle and nuclear physics. As the only electrically neutral fermions in the Standard Model (SM) of particle physics, neutrinos are the sole SM candidates for possessing a Majorana mass. The corresponding term in the Lagrangian breaks the conservation of total lepton number (L) by two units: L M ⊃ −y ν C H T H /Λ, where and H are the SM left handed lepton doublet and Higgs doublet, respectively, and Λ is a mass scale whose presence is needed to maintain dimensionality. After the neutral component of the Higgs doublet obtains a vacuum expectation value (vev) v/ √ 2 , the resulting Majorana mass operator is L M → −(m ν /2)ν c ν, with m ν = y ν v 2 /Λ. For y ν ∼ O(1), the observed scale of light neutrino masses consistent with oscillation experiments [1] and cosmological bounds [2,3] would imply Λ > ∼ 10 15 GeV. An experimental determination that neutrinos are Majorana fermions could, thus, provide circumstantial evidence for L-violating processes at ultra-high energy scales involving new particles not directly accessible in the laboratory. In the widely-considered see-saw mechanism, the L-violating, out-of-equilibrium decays of these particles (fermions) could generate the cosmic matter-antimatter asymmetry [4]. Neutrino oscillation experiments are agnostic regarding the existence of a Majorana neutrino mass term. However, the observation of 0νββ-decay in the nuclear transition (A, Z) → (A, Z + 2) + e − + e − [5] -a process that also violates L by two units -would provide conclusive evidence that light neutrinos are Majorana fermions [6].
The recent 0νββ-decay search in the KamLAND-Zen experiment [7] provides the most stringent upper limit on the effective Majorana mass |m ββ |, which is 0.061 − 0.165 eV at 90% confidence level (C.L.), where the range reflects the uncertainty in nuclear matrix element (NME) computations. In the three-neutrino framework [8], |m ββ | depends on the neutrino mass spectrum. In the inverted hierarchy (IH) it is bounded below |m ββ | > ∼ 0.01 eV, while in the normal hierarchy (NH) it can be vanishingly small. The next generation of 0νββdecay searches with ton-scale detectors [9][10][11][12][13][14] aim for sensitivities for |m ββ | as low as 0.01 eV. If neutrinos are Majorana fermions, and if the IH is realized in nature, one would thus expect a non-zero result in the ton-scale experiments.
Cosmological observations provide complementary information on neutrino masses, currently constraining the sum of neutrino masses (dubbed Σm ν ) to be smaller than 0.12 eV at the 2σ level [15]. Global fits [2,3] of neutrino oscillation data, 0νββ-decay search results, and cosmological surveys show that the NH is favored over the IH at about 2σ level. For future cosmological surveys [16][17][18][19][20], it is possible to exclude the IH, while the favored |m ββ | may be out the reach of ton-scale 0νββ-decay experiments [9][10][11][12][13][14] in the three-neutrino framework. Then, it is natural to ask how one could interpret a 0νββ-decay signal if cosmological measurements and/or future oscillation experiments demonstrate conclusively that the light neutrino mass ordering is in the NH.
Here, we address this question in the context of one of the most extensively studied extensions of the SM that generically implies the existence of Majorana neutrinos: the minimal left-right symmetric model (mLRSM) [21][22][23][24][25][26] light neutrino contribution if Λ 3.7 (5.9) TeV. In particular, it has been shown [31] that in the mLRSM the contributions coming from heavy neutrinos from the exchange of two right-handed W R bosons (the RR amplitude), see Fig. 1(a), are sizable. Nonetheless, the bulk of the mLRSM parameter space would remain largely inaccessible to ton-scale 0νββ-decay searches if cosmological data push the bound on Σm ν below ∼ 0.1 eV.
In what follows, we show that this conclusion changes dramatically in the presence of mixing between the leftand right-handed gauge bosons. This mixing results in contributions to the decay amplitude involving the exchange of heavy right-handed neutrinos, one SM W boson (predominantly left-handed) and one heavy W boson (predominantly right-handed) -a contribution we denote as the LR amplitude, see Fig. 1(d). In Ref. [32,33] it was found that the LR amplitude is suppressed with respect to the RR amplitude due to the upper bounds on the W L -W R mixing angle. However, those studies did not include long-range contributions associated with pion exchange that significantly enhance LR amplitude and can compensate for this suppression [34]. In this Letter, we compute these long-range contributions using state-ofthe art information on hadronic and nuclear matrix elements as well as phenomenological constraints on the relevant mLRSM parameters. We find that even in the presence of prospective, stringent cosmological bounds on Σm ν and possible exclusion of the IH, there exists ample opportunity for the observation of a signal in next generation 0νββ-decay searches.
This framework entails extending the SM gauge group to and L denote the SM abelian baryon and lepton quantum numbers. The Higgs sector consists of two scalar triplets ∆ L ∈ (1, 3, 2), ∆ R ∈ (3, 1, 2) and one bidoublet Φ ∈ (2, 2, 0), where (X, Y, Z) denote the representations under the SU(2) R,L and U(1) B−L groups. The neutral components of the bidoublet field Φ obtain vevs: and α being the spontaneous CP-violating phase.
Of particular relevance to 0νββ-decay is the chargedcurrent Lagrangian where A = L, R and V CKM L,R and V L,R are the Cabibo-Kobayashi-Maskawa (CKM) and lepton-mixing matrices, respectively. The L, R gauge bosons in terms of the light and heavy mass eigenstates W 1 and W 2 are given by W +µ [35], implying λ < 3.4 × 10 −4 . Tests of CKM unitarity place constraints on ξ. From recent results for the radiative corrections to nuclear βdecay [36,37], 0.25 × 10 −3 ≤ ξ ≤ 1.25 × 10 −3 is allowed at 95% C.L. in order to restore the CKM unitarity. We will consider the range 0 ≤ ξ ≤ 1.25 × 10 −3 . If the LR symmetry is taken to be parity (P), | sin α tan(2β)| < 2m b /m t [38,39] with m b and m t being the bottom and top quark masses, respectively. No such constraint exists when charge conjugation (C) is the LR symmetry [38,39]. In Ref. [40], they also consider the LR symmetry as C but with the maximum of tan β being m b /m t . Constraints from kaon CP violation and neutron electric dipole moments apply when α = 0 [41][42][43][44][45][46]. Here we consider C as the LR symmetry and α = 0 since our results are rather insensitive to fundamental sources of CP violation. There is no direct experiment bound on tan β so we choose tan β < 0.5 to keep the bidoublet Yukawa coupling of order unity. We will assume M W R = 7 TeV, which satisfies all the aforementioned constraints as well as the requirement of M W R > ∼ 6 TeV from the renormalization group evolution (RGE) analysis [47].
For purposes of illustration, we follow Ref. [31] and assume "type-II dominance" for neutrino masses 1 . In this scenario, m Ni ∝ m νi , one has V L = V * R [31]. Using the light neutrino mass difference from solar and atmospheric neutrinos [48], M W R = gv R , and fixing the neutrino mass m Nmax (= m N3 for the NH and = m N2 for the IH), it is possible to obtain all the neutrino masses in terms of the lightest neutrino mass m νmin .
The effective Lagrangian below the electroweak scale is where [34] and α, β are the color indices, τ ± = (τ 1 ± τ 2 )/2, τ 1 and τ 2 are the Pauli matrices. Wilson coefficients C 3R , C 3L and C 1 are obtained by integrating out the W 1,2 and N i arising respectively from the amplitudes in Fig. 1(a) gives [40,50,51] where C 1 (M W1 ) = 0 and it appears due to the RGE of C 1 . In Eq. (6), the non-vanishing Wilson coeffcients at the electroweak scale are given by are matched to effective operators above the electroweak scale, which however do not evolve under QCD running [40], so that the RGE only includes step (b).
The doubly charged scalar, depicted in Fig. 1(b), contributes solely to C 3R . When the LR symmetry holds, this contribution is negligible due to collider bounds [52] and charged lepton flavor violation constraints [31]. On the contrary, when the LR symmetry is explicitly broken, these constraints are relaxed and the corresponding contribution to the 0νββ-decay rate can be appreciable. For a discussion, and the possible interplay with prospective future low-and high-energy probes, see Ref. [53]. Here, we assume a LR-symmetric Lagrangian and leave the analysis of the interesting case when it is broken for a future work.
We now map the operators in Eq.
(2) at GeV scale ∼ Λ H onto an effective hadron-lepton Lagrangian below that scale [34,40,54] using chiral perturbation theory (χPT) [55,56]. Matching entails identifying all operators at a given chiral order that transform under chiral SU(2) the same way as the four-quark factor of a given operator in Eq. (2) [34,57]. We refer the reader to Ref. [34] for a detailed derivation, and here simply quote the results.
The hadron-lepton Lagrangian for the ππēe c ,N N πēe c andN NN Nēe c operators up to NNLO in chiral expansion is [34] The first two-pion term contributes to the amplitude A(nn → ppe − e − ) at order of p −2 with p m π being the typical momentum transfer. When this leading-order (LO) amplitude A LO is present as in the mLRSM, it can give a dominant long-range contribution to the halflife of 0νββ-decay [34]. The one-pion and four-nucleon and another two-pion terms, however, contribute at nextto-next-to LO (NNLO) to the amplitude A NNLO ∼ p 0 . The dimensionless coefficients are expressed as [34]  The four-nucleon interaction in Eq. (7) merits a more detailed discussion. In Ref. [40], it was observed that a consistent renormalization of the amplitude induced by the operators O ++ 1+ , O ++ 1+ requires inclusion of a LO four-nucleon counterterm [60]. While its presence does not change the magnitude of the NNLO contributions (barring accidental cancellations), it does introduce additional hadronic uncertainties at LO. To check how this new source of uncertainty might affect our results, we have taken the natural assumption that this new contribution gives an additional 100% contribution to the decay rate and found that our conclusions remain the same. Finally and notwithstanding the above arguments, the uncertainty might be bigger, as suggested by the RGE analysis of Ref. [40]. However, this issue is still far from settled until the finite piece of the LO four-nucleon counterterm is taken from a more reliable source, such as lattice QCD for instance.
From Eq. (7), we obtain the decay half-life  [48], and the Majorana phases are marginalized. The allowed regions with tan β = 0 and tan β ≤ 0.5 are depicted in darker and lighter colors. Red (green) dots denote the NH (IH) of neutrino mass ordering. Gray and orange lines represent the current and expected limits limits from the KamLAND-Zen [7] and future ton-scale experiment [9,10], respectively. The lightest heavy neutrino mass is also given in the upper horizontal axis.
where m ee and |m ee with m N = 939 MeV and Future ton-scale experiments searching for 0νββ-decay in 136 Xe are considered for numerical results. The phase space factor G −1 0ν = 7.11 × 10 24 eV 2 · yr [61,62], and the nuclear matrix elements In Fig. 2, we show the effective Majorana mass |m ββ | as a function of m νmin with m Nmax = 500 GeV and M W R = 7 TeV. To illustrate the impact of the LR contribution, we give the allowed regions with tan β = 0 (studied in Refs. [31,64]) and 0 < tan β ≤ 0.5 in darker and lighter colors, respectively. For most of the tan β > 0 parameter space, the long-range pion exchange contribution dominates over other contributions. In Fig. 3, we plot the |m ββ | as a function of m ν along with the current upper bound from cosmology experiments [15]. In particular, we see from Fig. 3 (upper panel) that in the NH, inclusion of the long-range contribution opens up a significant portion of parameter space accessible to tonscale experiments. Thus, even if the future CMB and LSS data would exclude the IH [16], there are good prospects of new physics at the TeV scale giving the dominant contribution to the 0νββ-decay rate in future ton-scale ex-periments.

ACKNOWLEDGEMENTS
GL and JCV would like to thank Jordy de Vries for many valuable discussions. GL thanks Wouter Dekens and Jordy de Vries for fruitful discussions on the half-life calculations regarding Refs. [40,65], Xiao-Dong Ma and Jiang-Hao Yu for the discussions on RGEs. JCV was supported in part under the U.S. Department of Energy contract DE-SC0015376. GL and MJRM were supported in part under U.S. Department of Energy contract DE-SC0011095. MJRM was also supported in part under National Science Foundation of China grant No. 19Z103010239.