New physics effects on neutrinoless double beta decay from right-handed current

We study the impact of new physics contributions to neutrinoless double beta decay arising from right-handed current in comparison with the standard mechanism. If the light neutrinos obtain their masses from Type-II seesaw within left-right symmetric model, where the Type-I contribution is suppressed to negligible extent, the right-handed PMNS matrix is the same as its left-handed counterpart, making it highly predictable and can be tested at next-generation experiments. It is very attractive, especially with recent cosmological constraint favoring the normal hierarchy under which the neutrinoless double beta decay is too small to be observed unless new physics appears as indicated by the recent diboson excess observed at ATLAS. The relative contributions from left- and right-handed currents can be reconstructed with the ratio between lifetimes of two different isotopes as well as the ratio of nuclear matrix elements. In this way, the theoretical uncertainties in the calculation of nuclear matrix elements can be essentially avoided. We also discuss the interplay of neutrinoless double beta decay measurements with cosmology, beta decay, and neutrino oscillation.

We study the impact of new physics contributions to neutrinoless double beta decay arising from right-handed current in comparison with the standard mechanism. If the light neutrinos obtain their masses from Type-II seesaw within left-right symmetric model, where the Type-I contribution is suppressed to negligible extent, the right-handed PMNS matrix is the same as its left-handed counterpart, making it highly predictable and can be tested at next-generation experiments. It is very attractive, especially with recent cosmological constraint favoring the normal hierarchy under which the neutrinoless double beta decay is too small to be observed unless new physics appears as indicated by the recent diboson excess observed at ATLAS. The relative contributions from left-and right-handed currents can be reconstructed with the ratio between lifetimes of two different isotopes as well as the ratio of nuclear matrix elements. In this way, the theoretical uncertainties in the calculation of nuclear matrix elements can be essentially avoided. We also discuss the interplay of neutrinoless double beta decay measurements with cosmology, beta decay, and neutrino oscillation.

I. INTRODUCTION
Although the Standard Model (SM) of particle physics is extremely successful in explaining most of the experimental data up to current accelerator energies, it does not explain non-zero neutrino masses confirmed by solar, atmospheric and reactor oscillation data, as well as the mystery of dark matter and the matter dominance of present universe. These are the experimentally facts which motivate physics beyond the standard model and in addition there exist various theoretical reasons.
Left-Right Symmetric Models (LRSM) [1][2][3][4][5][6] are well motivated candidates of physics beyond SM. Among the reasons are: i) explaining light neutrino masses via seesaw mechanism with the natural embedding of righthanded neutrinos, ii) providing theoretical origin of maximal parity violation seen at weak interaction while conserved in strong and electromagnetic interaction, iii) the fact that B − L is a more attractive U(1) quantum number and the symmetry of U(1) charges of the fermions. With the scalar sector comprising of Higgs bidoublet and triplets, the light neutrino mass is generated by type-I plus type-II seesaw mechanism. The Higgs scalars generate masses for light and heavy neutrinos which not only accommodates the Majorana nature of light and heavy neutrinos, but also leads to testable lepton number violation (LNV) at low energy experiments like neutrinoless double beta decay (0νββ) and at high energy experiments like the Large Hadron Collider (LHC, see refs. [7] for a detailed discussions). The LHC can also probe right-handed current. Recently ATLAS [8] claims to find a diboson excess which can be explained by a 2 TeV W via pp → W → W Z, and CMS [9][10][11] sets stringent lower bound on the W mass through pp → W → +2j. In the context of LRSM, in principle, one can derive lim-its for right-handed neutrino mass and mixing, W R mass and its mixing with the left-handed counterpart as well as the corresponding gauge coupling g R .
On the other hand, neutrinoless double beta decay is a unique phenomena whose experimental observation would reveal the character of light neutrinos i.e, whether they have Majorana mass term [12] which violates lepton number. If so, neutrinoless double beta decay can happen. At present, the best lower limit on the decay half-life using 76 Ge is T 0ν 1/2 > 2.1 × 10 25 yrs at 90% C.L. from GERDA [13] while the combined bound is T 0ν 1/2 > 3.0 × 10 25 yrs. At the same time, the future experiment GERDA Phase-II [14] is expected to improve the half-life sensitivity to reach T 0ν 1/2 > 2.0×10 26 yrs. For 136 Xe, the derived lower limits on half-life from EXO-200 and KamLAND-Zen experiments are T 0ν 1/2 > 1.6 × 10 25 yrs [15] and T 0ν 1/2 > 1.9 × 10 26 yrs [16], respectively. The combined limit from KamLAND-Zen Collaboration is T 0ν 1/2 > 3.4 × 10 26 yrs at 90% C.L. Lepton number violating 0νββ transition could be induced by either light Majorana neutrinos or new physics. For the latter, one has to go beyond SM and many models can contribute to neutrinoless double beta decay [7,[17][18][19][20][21][22][23][24][25][26][27][28][29][30]. The nest-generation experiments of neutrinoless double decay can touch down to the lower limit of inverted hierarchy (IH), corresponding to effective Majorana mass parameter m ν ee around 0.01 eV. Nevertheless, the latest cosmological bound on the mass sum of light active neutrinos [31] has indicated that the normal hierarchy (NH) is favored at 1σ confidence level with tiny smallest mass. A direct consequence is that neutrinoless double beta decay is difficult to be observed at nextgeneration experiments if its only source comes from the standard mechanism. If the result of observation turns out to be the opposite, namely neutrinoless double beta decay is observed but cannot be explained by the standard mechanism, it clearly indicates physics beyond SM.
With these motivations, both hints from the diboson excess observed at ATLAS for heavy W around 2 TeV and cosmological preference for NH with tiny mass scale and invisible neutrinoless double beta decay even at nextgeneration experiments, we examine the effect of Type-II seesaw contributions to neutrinoless double beta decay arising from purely right-handed current within LRSM. The predicted effect mass can saturate the current experimental bounds making it testable in the future even with NH being favored.
The paper is organized as follows: After briefly summarizing the standard mechanism of neutrinoless double beta decay in Sec. II we consider new physics effect from purely right-handed current in Sec. III. Next we compare the new physics contribution with the standard mechanism and derive lower limit on the absolute scale of light neutrinos from current experimental limit in Sec. IV. In Sec. V, we use half-life and nuclear matrix element (NME) ratios to reconstruct the relative contributions of the standard mechanism and new physics to effective Majorana mass parameter. This approach to distinguishing different sources has a significant advantage that the theoretical uncertainties in the calculation of NME can be essentially avoided. We conclude our paper in Sec. VI.

II. NEUTRINOLESS DOUBLE BETA DECAY
Observing neutrinoless double beta decay can not only reveal LNV but also provide crucial information about the absolute scale of light neutrino masses and the associated mass-generation mechanism. If light Majorana neutrinos are the only source of 0νββ transition, we can express the half-life as which contains three important factors: i) the NME M 0ν ν , ii) the phase space factor G 0ν 01 , and iii) a dimensionless particle physics parameter -a measure of lepton number violation η ν ≡ m ν ee /m e which is a function of neutrino masses, mixing angles, and CP phases, where m i are the mass eigenvalues of light neutrinos, α, β are the two Majorana CP phases, and U ei are the elements of PMNS mixing matrix. We present the variation of |m ν ee |, due to the unknown Majorana CP phases, as a function of the lightest neutrino mass (m 1 for NH and m 3 for IH) in Fig.1. The notation of mixing angles is defined as (c α , s α ) ≡ (cos θ α , sin θ α ) where θ a ≡ θ 23 is the atmospheric mixing angle, θ r ≡ θ 13 the reactor mixing angle, and θ s ≡ θ 12 the solar mixing angle. In addition, the oscillation data are sensitive to two mass-squared differences, the solar mass-squared difference ∆m 2 s ≡ m 2 2 − m 2 1 and the atmospheric mass-squared difference ∆m 2 a ≡ |m 2 3 − m 2 1 |. Since we know only the magnitude of the atmospheric mass-squared difference with its sign unknown, there exit two different mass patterns among light neutrinos: NH with m 1 < m 2 < m 3 and IH with m 3 < m 1 < m 2 . The 3σ global fit ranges for the oscillation parameters are summarized in Table I. The effective Majorana mass parameter of neutrinoless double beta decay that predicted by the standard mechanism is displayed in Fig. 1 with light-red (green) band for NH (IH), respectively. The horizontal bands are bounds from neutrinoless double beta decay measurements which can provide crucial information about the absolute scale of light neutrino mass and Majorana CP phases. In addition, the mass scale can also be bounded from i) the measurement of beta decay parameter m β = i U 2 ei m i for which the existing bound is m β < 0.2 eV from KA-TRIN [33], ii) The limits on the effective Majorana mass parameter from various isotopes [30], iii) Cosmological observations which provide constraints on the sum of light neutrino masses m Σ ≡ i m i . They have been shown as vertical bands in Fig. 1. Note that these constraints have only touched down to the quasi-degenerate region, m 1 m 2 m 3 and m lightest > ∆m 2 a , with the beta-decay and cosmological constraints approximately at m β ≥ |m ν ee | which is the same for both hierarchies. The current bound on the sum of light neutrino mass is m Σ < 0.23 eV derived from Planck+WP+highL+BAO data (Planck1) at 95% C.L. while m Σ < 1.08 eV from Planck+WP+highL (Planck2) at 95% C.L. [34]. It is claimed that the latest cosmological constraint can be approximated by gaussian distribution, m Σ = 22 ± 62 meV [31]. At 1σ C.L., the mass sum is smaller than 84 meV which is below the smallest value for IH. In this sense, the NH is favored. There is no hope to probe the normal hierarchy within the standard mechanism, even for nextgeneration experiments. On the contrary, if there is new physics contribution in addition to the standard mechanism, the neutrinoless double beta decay can be sizable. In the following part, we discuss one possibility of LRSM with Type-II seesaw dominance which is fully predictable and can be tested at next-generation experiments.   [32].

III. EFFECT OF NEW PHYSICS CONTRIBUTION
In this section we briefly discuss a simple left-right symmetric model as the source of new physics that can contribute to neutrinoless double beta decay and is fully predictable. The left-right symmetric model [1][2][3][4][5][6] is based on the gauge group SU under parity. The quarks and leptons individually transform as Spontaneous symmetry breaking is implemented as usual with the help of triplet and bidoublet Higgs fields, with quantum numbers (3, 1, 2), (1, 3, 2), and (2, 2, 0), respectively. The bidoublet connects left-and right-handed fermions to provide Dirac mass term while the triplet provides Majorana mass term to left-or right-handed neutrinos. After symmetry breaking, the neutrino mass matrix can be written as, where, is the LH (RH) Majorana masses for light (heavy) neutrinos while m D = yv 1 +ỹv 2 is the Dirac neutrino mass term connecting light and heavy neutrinos. The parity is first broken by v R , the vacuum expectation value (VEV) of the right-handed triplet ∆ R , reducing The Majorana nature of the light and heavy neutrinos, arising from the presence of the scalar triplets ∆ L,R with charge B − L = 2, implies that they can mediate neutrinoless double beta decay [7,[24][25][26][27][28][29], as shown in Fig. 2. The left-and right-handed currents contribute independently without interference. The other diagrams contributing to neutrinoless double beta decay via doubly charged scalar triplet exchange, λ-diagram involving W L − W R mediation and η-diagram are suppressed, for details, see refs. [25,28,29].
A. Type-II seesaw dominance As shown in (5), the light neutrino mass matrix receives two independent contributions from Type-I and Type-II seesaw mechanisms. Although the left-and right-handed Yukawa/mass matrices are connected with each other, M L ∝ M R , due to parity, the Dirac mass matrix m D is independent of the others. Experimentally, we have measured the oscillation parameters of the light neutrinos. The information is not enough to constrain both M L (M R ) and m D . While the light neutrino mass matrix m ν can be reconstructed, the heavy one M R is out of our knowledge. On the other hand, both the light and heavy neutrino mass matrices are involved in the neutrinoless double beta decay process as shown in Fig. 2. It is difficult to make a specific prediction unless we can somehow know the heavy neutrino mass matrix M R . Fortunately, Type-II seesaw dominance can save the situation here [7,25,27,28,35].
With Type-II seesaw dominance, which can be realized by suppressing the Dirac mass term m D , the mass matrices of the left-and right-handed neutrinos are proportional to each other, A direct consequence is that the left-and right-handed neutrinos share the same mixing matrix, in the diagonal basis of charged leptons. The unmeasured right-handed neutrino mixing can then be fully reconstructed. Actually, the equality between the left-and righthanded PMNS mixing matrices can also come from the lepton sector. To see this let us first write down the Yukawa couplings that are responsible for the lepton mass matrix, where Φ † ≡ σ 2 Φ * σ 2 is the CP mirror of the bidoublet Φ with τ 2 being the second Pauli matrix. When the Higgs bidoublet develops a non-zero VEV, the leptons receive a mass matrix, If the Higgs doublet fails to develop trivial CP phases or if the two Yukawa matrices Y and Y commute with each other, the lepton mass matrix can commute with its Hermitian conjugate, A direct consequence is that the left-and right-handed leptons share the same mixing matrix, If the neutrino mixing is trivial, V ν = I, then the PMNS mixing matrices come solely from the charged lepton mixing, reproducing (7). For all cases, the mass eigenvalues for light as well as heavy neutrinos are related as, where M i and m i are physical masses for light active and heavy Majorana neutrinos. Fixing M max ( M 3 for NH and M 2 for IH), the mass relation can be expressed as,

IV. COMPARISON BETWEEN NEW PHYSICS AND STANDARD MECHANISM
The inverse of half-life for a particular isotope, with new physics effects realized via a left-right symmetric model assuming Type-II seesaw dominance, contributes to neutrinoless double beta decay therefore in the following way, where m    4 contains the typical momentum transfer p ≈ 100MeV. Here, the lepton number violating particle physics parameters are η ν and η N (m ν ee , m N ee ), while G 0ν 01 is the phase space factor, and M 0ν ν (M 0ν N ) the NMEs derived for virtual light and heavy particle exchanged diagram, respectively, whose values can be found in Table II. The SU (2) L and SU (2) R coupling constants g L and g R , respectively, need not to be the same for LRSM. In addition to an overall factor C N /M 3 for NH or C N /M 2 for IH, m N ee share the same set of input parameters as m ν ee . To estimate the lepton number violating parameter arising from righthanded current, It is necessary to know oscillation parameters, the Majorana CP phases, and the mass scales.
• Oscillation parameters: Neutrino oscillation experiments has measured the two mass-squared differences and the three mixing angles. Nevertheless only two mixing angles, the reactor mixing angle θ r and the solar mixing angle θ s , are relevant in the effective Majorana mass parameter. The error of these four oscillation parameters can introduce uncertainty in the prediction of |m ν ee |, especially the solar mixing angle θ s [37]. With the current global fit [32], the upper limit on the minimal half-life for IH varies by a factor of 2.5 at 3σ C.L. Fortunately, the same set of oscillation parameters is shared by reactor neutrino experiments and hence can be precisely measured by next-generation medium-baseline experiments like JUNO [38]. The uncertainty from oscillation parameters can be reduced to below 1% which is essentially negligible [30].
• CP Phases: Although the Dirac CP phase δ D also enters the effective Majorana mass parameter |m ν ee | and |m N ee |, it will not manifest itself in the predicted values. This is because it is always associated with one of the two Majorana CP phases α and β which are completely unknown. For simplicity, the Dirac CP phase has been omitted in the expressions. The variation of α and β renders the predicted values of |m ν ee | and |m N ee | to span in a wide band. In Appendix A, we generalize the geometrical picture of obtaining the minimal and maximal values within standard mechanism to LRSM-Type II seesaw dominance.
• Mass Scales: In addition to oscillation parameters and CP phases, the mass scales of light and heavy neutrinos are still unknown yet. For light neutrinos, we fix the mass scale by hand since there is no measurement at all. On the other hand, the contribution from heavy neutrinos involve two mass scales, the heavy neutrino and W R masses. First, we need to fix the heavy neutrino mass scales by fixing M 3 for NH or M 2 for IH. Second, the W R mass also enters through C N . When combined, the net effect is an overall factor, 1/(M 4 W R M 3 ) for NH or 1/(M 4 W R M 2 ) for IH. A natural realization is M 2,3 should be roughly of the same scale as M W R . In other words, the low-energy neutrino phenomenology is highly related with collider signature. If the di-boson excess [8] observed at ATLAS really comes from LRSM with mass M W R ≈ 2 TeV, its contribution to low energy neutrino phenomenology is large enough to be observed by next-generation neutrinoless double beta decay experiments.
We show in Fig. 3 the lepton number violating effective Majorana mass parameters vs lightest neutrino mass for 76 Ge which is used by GERDA [13]. For comparison, we show both the total contribution and the one from standard mechanism without new physics. The numerical analysis has been performed using 3σ prior and posterior distributions from current global fit and the one improved with JUNO [38]. As studied in [30] the combination of short-and medium-baseline reactor experiments can achieve precise measurement of the oscillation parameters involved in neutrinoless double beta decay. Here, we adopt the same configuration of JUNO with 53 km of baseline, 20 kt of detector, 36 GW of thermal power, 6 years of running (300 effective days for each year), and energy resolution 3%/ E/MeV. This roughly reproduce the results of the JUNO yellow book [38]. It is clearly shown in Fig. 3 that the uncertainty from oscillation parameters has significant effect on the prediction of the effective Majorana mass parameter within not only the standard mechanism but also the total contribution. In Fig.3 we have fixed M W R 2 TeV, g R ≈ 2/3g L , and M N = 1 TeV which can satisfy both ATLAS [8] and CMS [9] excesses. With this typical mass assignment indicated by collider signature, the contribution from new physics is not small and actually can dominate over the standard mechanism in low mass region. In the quasi-degenerate region, there is no big difference between the total contribution and the standard mechaism. As the mass scale of light neutrinos decreases, the new physics contribution keeps growing and stops the total one from decreasing to zero for both hierarchies. The minimal value is within the sensitivity region of next-generation neutrinoless double beta decay experiments which can touch down to around 0.01 eV. When the smallest mass eigenvalue m 1,3 of the light neutrinos further decreases, the new physics contribution dominates and can finally saturate the current experimental bound placed by GERDA [13], Heidelberg-Moscow [39], and IGEX [40]. Since no experiment has observed neutrinoless double beta decay, we can derive lower limit on the absolute scale of light neutrinos, m 1 > 2.3 meV for NH , (17a) m 3 > 0.075 meV for IH .
(17b) as indicated by solid vertical lines in Fig. 3. This is totally different from the standard mechanism based on which neutrinoless double beta decay experiments can only provide upper limits. For IH, the standard mechanism cannot provide lower limit which may be possible for NH. On the contrary, the LRSM-Type II contribution can place both lower and upper limits on the mass scale of light neutrinos.
The lower limits obtained from current neutrinoless double beta decay measurements are consistent with the cosmological constraint on the sum of light neutrino masses [41][42][43], For clarity, we replot the effective Majorana mass parameter in Fig. 4 as a fuction of the mass sum. It clearly shows that at more than 1σ C.L., NH is favored over IH. If so, the low-energy neutrinoless double beta decay cannot be observed even at next-generation experiments, if no new physics is present. Fortunately, the diboson excess observed at ATLAS provides a timely way out.

V. SEPARATING THE SM AND LRSM CONTRIBUTIONS
In LRSM with Type-II seesaw, the two contributions are linearly combined as shown in (15). Using two isotopes, (A i , Z i ) and (A j , Z j ), it is possible to separate the SM and non-interfering new physics contributions [36,44], However, there is sizable uncertainty in the theoretical calculation of the nuclear matrix elements which implies The standard mechanism contribution is plotted with hatched bands, the dense blue one for NH and sparse yellow one for IH, while the new physics contribution with shaded bands, the blue one for LR-NH and the red one for LR-IH. The comparison between SM and LRSM-Type II seesaw contributions to the effective Majorana mass parameter |mee| as a function of the mass sum has been presented using JUNO data and bound from cosmology. The limits on mass sum at 1σ, 2σ and 3σ C.L., mΣ < 84 meV, mΣ < 146 meV, and mΣ < 208 meV, respectively, are shown as vertical lines.  Fig. 3. In the right, we show the ratio |m N ee /m ν ee | as a function of the mass sum mΣ together with recent cosmological constraints. Here |m ν ee | is the standard mechanism contribution with WL-WL mediation while |m N ee | the LRSM-Type-II seesaw contribution via WR-WR mediation.
that the extracted effective Majorana mass parameters |m ν ee | and |m N ee | have uncertainties.
Fortunately, the theoretical uncertainties can be significantly reduced by taking the ratios between nuclear matrix elements as well as between half-lifes of different isotopes. These ratios can be used to obtain the ratio of new physics and standard mechanism effective Majorana mass parameters, One can therefore expect that this ratio will be more precisely determined than the individual effective Majorana mass parameters.
In Fig. 5, we show the ratio |m N ee /m ν ee | as a function of the lightest neutrino mass in the left panel. Its value can span several orders of magnitude. In the quasidegenerate region, the ratio is typically below 0.1. For NH, it grows with decreasing mass scale. On the other hand it can experience a short range, 1 meV m 3 4 meV, of touching down to the bottom, although the maximal value keeps growing with decreasing m 3 . Across the whole range, the ratio for NH is always larger than its value for IH. Together with the lower limit on the lightest mass extracted from current neutrinoless double beta decay measurements, as depicted in Fig. 3, the ratio is constrained to be smaller than 10 3 for IH while it can go well above 10 4 for NH. If neutrinoless double beta decay are observed with two isotopes at future experiments to reconstruct the ratio |m N ee /m ν ee |, it is possible to determine the neutrino mass hierarchy.
According to the definition (16) of effective Majorana mass parameters, the contribution of the right-handed sector actually can be separated from light neutrino parameters since it only appears as an overall factor C N /M 3 for NH or C N /M 2 for IH. From the measured ratio (20) we can reconstruct, which can be compared with collider measurements on the right-handed sector parameters contained in M 2,3 /C N . This can be used to add another input to determine the Majorana CP phases. The degeneracy between α and β can then be eliminated.

VI. CONCLUSIONS
We have shown that new physics contributions to neutrinoless double beta decay induced by right-handed current can saturate the present experimental bound. Then, the neutrinoless double beta decay experiment is still possible to see a signal for NH which together with small mass scale is favored by the recent bound on m Σ from cosmology. In comparison, if only the standard mechanism contributes, it is difficult to see a signal of neutrinoless double beta decay even at next-generation experiments. Fortunately, collider signature from ATLAS and CMS has indicated heavy gauge boson W around 2 TeV, pointing to a timely way out.
We studied a simple framework of LRSM to provide the new physics effects to neutrinoless double beta decay. The neutrino mass mechanism is governed by Type-II seesaw dominance such that the mass eigenvalues and mixing matrices of light and heavy neutrinos are correlated with each other. To illustrate the idea, we carried out numerical estimation for M W R ≈ 2 TeV in order to show the typical distribution.
We derived the corresponding lower limits on the absolute scale of light neutrinos, m 1 2.3 meV for NH and m 3 0.075 meV for IH by saturating the limits set by current neutrinoless double beta decay experiments. Different from the standard mechanism, where only upper limit on the mass scale can be extracted from neutrinoless double beta decay measurements, LRSM-Type II seesaw allows both upper and lower limits. In addition, the total effective Majorana mass parameter is bounded from below and well within the sensitivity reach of the nextgeneration experiments. To distinguish the two noninterfering contributions from the light and heavy neutrinos, we take ratios between the half-lifes and nuclear matrix elements for two isotopes. In this way, the theoretical uncertainties in the calculation of nuclear matrix elements can be avoided. Once measured and LRSM-Type II seesaw is confirmed, the ratio |m N ee /m ν ee | can be interpreted to distinguish the light neutrino mass hierarchy. Further, supplemented with collider measurements of the right-handed sector parameters, the two Majorana CP phases α and β can be uniquely determined without degeneracy. m N ee 2 ≡ |F 1 + F 2 e iα + F 3 e iβ | 2 . (A2b) When expanded, they can be written in terms of real functions, |m ν ee | 2 = i f 2 i + 2f 1 f 2 cos α + 2f 1 f 3 cos β + 2f 2 f 3 cos(α − β) , (A3a) Both contributions share the same functional form of α and β. After combination, the functional form remains, with, The redefined elements G i can be readily solved, The combined effective electron mass can be formulated as, Then, the geometrical picture of finding the minimal and maximal values by varying α and β as elaborated in [45] can readily apply to the last term in (A7).