Neutrinoless double beta decay in the left-right symmetric models for linear seesaw

In a class of left-right symmetric models for linear seesaw, a neutrinoless double beta decay induced by the left- and right-handed charged currents together will only depend on the breaking details of left-right and electroweak symmetries. This neutrinoless double beta decay can reach the experimental sensitivities if the right-handed charged gauge boson is below the 100 TeV scale.


I. INTRODUCTION
The phenomena of neutrino oscillations have been established by atmospheric and solar neutrino measurements, and also confirmed by accelerator and reactor neutrino experiments [1]. This fact implies the need for massive and mixing neutrinos and hence new physics beyond the SU (3) c ×SU (2) L ×U (1) Y standard model (SM). Furthermore, the cosmological observations have indicated that the neutrino masses should be in a sub-eV range [1]. The seesaw [2][3][4][5] mechanism provides a natural way to the tiny neutrino masses [2][3][4][5][6][7][8][9][10][11][12]. For realizing a seesaw scenario, one can simply extend the SM by introducing some heavy particles with lepton number violation of two units. Another appealing scheme is to consider the SU (3) c ×SU (2) L ×SU (2) R ×U (1) B−L left-right symmetric models (LRSMs) [13][14][15][16], where the lepton number violation and the heavy particles can naturally appear after the left-right symmetry is spontaneously broken down to the electroweak symmetry. For example, the LRSMs with [SU (2)]-triplet and bidoublet Higgs scalars can accommodate a type-I+II seesaw [5,10].
It should be noted the neutrinos are neutral and hence are allowed to have a Majorana nature [17]. One thus can expect a neutrinoless double beta decay (0νββ) [18] process mediated by the Majorana electron neutrinos. This 0νββ process is determined by the Majorana neutrino mass matrix [19,20]. Alternatively, the 0νββ process can come from other lepton number violating interactions. For example, there have been a number of people studying the 0νββ processes in the TeV-scale LRSMs for the type-I+II seesaw [5,10,[21][22][23][24][25][26][27][28][29][30][31]. Specifically, people need do some assumptions on the scale of the righthanded neutrinos as well as the mixing between the leftand right-handed neutrinos to quantitatively analyze the 0νββ processes involving the right-handed charged currents. One may consider other models which induce a testable 0νββ process at tree level and then give a negligible contribution to the neutrino masses at loop level [32]. The 0νββ processes thus can be free of the neutrino masses [33][34][35]. But these models contain quite a few arbitrary parameters. * Electronic address: peihong.gu@sjtu.edu.cn In this paper we shall study the 0νββ processes in a class of LRSMs for the so-called linear seesaw [36,37], where the left-handed neutrinos have a Majorana mass matrix proportional to the Dirac mass term between the left-and right-handed neutrinos. Our illustration will show that a 0νββ process induced by the left-and righthanded charged currents together is irrelevant to any masses and mixing of the left-and right-handed neutrinos, instead it is only dependent on the vacuum expectation values (VEVs) of some Higgs scalars. This 0νββ process can arrive at a testable level if the right-handed charged gauge boson is below the 100 TeV scale.

II. LINEAR SEESAW
We start with the original LRSM, where the Higgs multiplets include two SU (2) doublets and one SU (2) L × SU (2) R bidoublet, Here and thereafter the brackets following the fields describe the transformations under the SU (3) c × SU (2) L × SU (2) R × U (1) B−L gauge groups. In the fermion sector, we have three generations of [SU (2)]-doublet fermions, One can analyze the symmetry breaking in details from the fully renormalizable scalar potential which contains The acceptable breaking pattern is SU Here the ϕ scalar is a linear combination of three SU (2) L doublets φ 1,2 and χ L , The allowed Yukawa interactions are L ⊃ −y qqL Φq R −ỹ qqLΦ q R −y llL Φl R −ỹ llLΦ l R +H.c., (5) from which one can read the fermion masses, Clearly one must fine tune some Yukawa couplings to make the Dirac neutrino masses below the eV scale. It has been shown that the original LRSM can generate the tiny neutrino masses in a natural way if it is extended by three gauge-singlet fermions [38,39], which have the Yukawa couplings, In principle, the singlet fermions are also allowed to have a gauge-invariant Majorana mass term. Here we simply assume that this term does not exist, i.e.
We will give a realistic model to explain this assumption in Sec. IV. Under this assumption, we can obtain three Dirac pairs composed of the right-handed neutrinos ν R and the singlet fermions ξ R at the electroweak level. By choosing the discrete left-right symmetry to be the CP under which the fermions and the scalars have the following transformations, one can simplify the Yukawa couplings (5) and (8) as The charged fermion mass matrices in Eq. (6) thus should be symmetric, i.e. m u = m T u , m d = m T d and m e = m T e . As for the neutral fermions, their masses are For f χ R ≫ m D and χ R ≫ χ L , the left-handed neutrinos can get a Majorana mass term through the seesaw mechanism, i.e. Remarkably, the Majorana mass matrix of the lefthanded neutrinos is proportional to the Dirac mass term between the left-and right-handed neutrinos. This formula of the neutrino masses is referred to as a linear seesaw. Usually one takes a sizable Dirac mass term m D and a small factor χ L / χ R to generate the tiny neutrino masses m ν . In the following we will show a small m D and a sizable χ L / χ R is interesting to the 0νββ process. In Sec. IV, we will also explain the small m D in a realistic model.

III. NEUTRINOLESS DOUBLE BETA DECAY
As shown in Fig. 1, the left-handed neutrinos ν L , the right-handed neutrinos ν R and the singlet fermions ξ R with the masses (12) can mediate some 0νββ processes in association with the charged currents, Here we have rotated the right-handed neutrinos and the singlet fermions to diagonalize their masses, i.e.
and hence have rewritten the mass matrix in Eq. (12) by By integrating out the heavy Dirac fermions composed ofν R andξ R , we can derive the effective 0νββ operators, (17) where the first and second terms (with q being the transfer momentum at the lepton vertex) are respectively from the diagrams (a)+(b) and the diagram (c) in Fig. 1.
Here we have ignored the mixing between the W L and W R gauge bosons for χ R ≫ ϕ . In this case the W L and W R masses are simply given by One can take the W L −W R mixing into account for more 0νββ processes. In general, a reasonably large W L − W R mixing can significantly contribute to the 0νββ processes. However, the W L −W R mixing can be as small as which is a one-loop contribution dominated by the top and bottom quarks. In this case, we can neglect the 0νββ processes from the W L −W R mixing. Actually, we will demonstrate a realistic model to explain the absence of the Majorana mass term of the singlet fermions as well as the smallness of the Dirac mass term between the leftand right-handed neutrinos. In this realistic model, the W L −W R mixing is dominated by Eq. (19).
The half-life of the 0νββ processes is calculated by with G 0ν 01 being the phase space factor and M 0ν LL,LR being the nuclear matrix elements. As the Yukawa couplings |y q(l) |, |ỹ q(l) | < √ 4π are allowed by the perturbation requirement, we can take χ L very close to ϕ , i.e. χ L ≃ ϕ . Accordingly the factor χ L / χ R can approximate to from the GERDA collaboration [43]. In the above demonstration, the mixing between the left-handed neutrinos and the singlet fermions is of the order of χ L / χ R ≃ M W L /M W R . Such mixing will affect the invisible decay width of the Z boson if the Dirac pairs composed of the singlet fermions and the righthanded neutrinos are heavier than the Z boson. Compared with its SM value, the Z invisible decay width will be reduced by a factor 1 − χ L 2 / χ R 2 . So, the present parameter choice can be consistent with the precise measurement Γ Z (invisible) = 499.0 ± 1.5 MeV [1] even if the singlet fermions and the right-handed neutrinos are heavier than the Z boson.

IV. A REALISTIC MODEL
The above demonstrations are based on two assumptions: (i) the absence of the Majorana mass term of the singlet fermions; (ii) the smallness of the Dirac mass term between the left-and right-handed neutrinos. We now present a realistic model to naturally account for these assumptions. Besides the previous fermions (2) and (7), this model contains other three generations of [SU (2)]singlet fermions, The Higgs sector (1) is also enlarged to be, χ La (1, 2, 1, −1), χ Ra (1, 1, 2, −1) (a = 1, 2). (22) There is a U (1) P Q global symmetry under which the fermions and the scalars carry the charges as shown in Table I. We will show later this U (1) P Q is indeed a Peccei-Quinn (PQ) symmetry [47][48][49] for solving the strong CP problem. Under the discrete CP symmetry and the global PQ symmetry, the scalar potential should include The gauge-singlet scalar σ is responsible for the spontaneous PQ symmetry breaking, i.e.
with f P Q = √ 2 σ being a VEV, h P Q being a Higgs boson and a being a Goldstone boson. Subsequently, we can define an [SU (2) R ]-doublet scalar and an [SU (2) L ]doublet scalar, to respectively drive the left-right and electroweak symmetry breaking. We would like to emphasize that the VEVs of the [SU (2) L × SU (2) R ]-bidoublet Higgs scalar Φ are seesaw-suppressed, Actually for χ L1,2 < ϕ = 174 GeV, it is easy to give φ 1,2 = O(1 − 100 eV) by inputting σ = O(10 10−12 GeV), M φ1,2 = O(10 12−13 GeV) and χ R1,2 = O(1 − 100 TeV). We can also take the dimensionless coefficients κ 11,12 andκ 22,12 small enough to suppress the VEVs φ 1,2 in the case the Higgs bidoublet Φ is at an accessible scale M φ1,2 = O(TeV) while the PQ symmetry is still constrained to break at a high scale σ O(10 10 GeV). The allowed Yukawa couplings are The fermion masses thus should be By block diagonalizing the above mass matrices, we have for the neutral fermions, and for the charged fermions. Compared with the fermion masses in the revived original LRSM, the neutrino masses are still induced by the linear seesaw, while the SM charged fermion masses additionally have a universal seesaw [50,51] contribution (the second terms in the charged fermion masses m u,d,e ). It is easy to find that for φ 1,2 = O(1 − 100 eV), χ L1,2 < ϕ = 174 GeV and χ R1,2 = O(1 − 100 TeV), the neutrino masses can naturally arrive at the sub-eV scale while the charged fermion masses should mostly come from the universal seesaw. The effective operators for the 0νββ processes are The half-life of the 0νββ processes can be computed by replacing the factor χ L 2 / χ R 2 to χ L2 2 / χ R2 2 in Eq. (20). As an example, we can take the seesaw condition χ R2 ≫ χ L2 to be χ R2 2 = 1000 χ L2 2 , and then perform T 0ν 1/2 ( 136 Xe) > 8×10 26 yr for M W R > 32.4−36.9 TeV and T 0ν 1/2 ( 76 Ge) > 6 × 10 27 yr for M W R > 32.5 − 47.6 TeV. With this parameter choice, the mixing between the left-handed neutrinos and the singlet fermions will not conflict with the precise measurement of the invisible decay width of the Z boson even if the Dirac pairs composed of the right-handed neutrinos and the singlet fermions are heavier then the Z boson. Alternatively, we can consider a little weaker seesaw condition χ R2 = 10 χ L2 by assuming the right-handed neutrinos and the singlet fermions much lighter than the Z boson. In this case, we can have T 0ν 1/2 ( 136 Xe) < 8 × 10 26 yr for M W R < 57.7 − 65.8 TeV and T 0ν 1/2 ( 76 Ge) < 6 × 10 27 yr for M W R < 57.8 − 84.7 TeV. The present model has some advantages. Firstly, it can automatically forbid the Majorana mass term of the singlet fermions and naturally suppress the Dirac mass term between the left-and right-handed neutrinos. We hence need not artificially take the essential assumptions in the previous demonstrations. Secondly, the light quarks u, d and the heavy quarks U, D have the axial couplings to the Goldstone boson a as follows, The Goldstone boson a will contribute to the strong CP phase through the DFSZ [52,53] scheme (the axial couplings to the light quarks u, d) and the KSVZ [54,55] scheme (the axial couplings to the heavy quarks U, D). Therefore, the Goldstone boson a is an invisible axion and the global symmetry U (1) P Q is a PQ symmetry.
The PQ symmetry breaking scale should have a lower limit f P Q 10 10 GeV to avoid the astrophysical constraints. The axion a, which eventually picks up a tiny mass through the color anomaly and then becomes a pseudo Goldstone, can act as a cold dark matter particle for a proper choice of the breaking scale of the PQ symmetry and the initial value of the strong CP phase [1]. Thirdly, the [SU (2) × SU (2) R ]-bidoublet Higgs scalar Φ has a seesaw-suppressed VEV. So its Yukawa couplings to the quarks can only give a negligible contribution to the quark masses and hence their values can be set flexibly. Meanwhile, the Yukawa couplings of this Higgs bidoublet to the leptons are completely determined by the neutrino mass matrix. The Higgs bidoublet is allowed near the TeV scale, so that it may be tested at the running and/or planning colliders. In particular the decays of the Higgs bidoublet into the charged leptons can open a window to measure the neutrino mass matrix.

V. SUMMARY
In this paper we have shown in the LRSMs for the linear seesaw, a 0νββ process induced by the left-and right-handed charged currents together decouples from any masses and mixing of the left-and right-handed neutrinos, but depends on the VEVs of the Higgs scalars for driving the left-right and electroweak symmetry breaking. This 0νββ process can reach the experimental sensitivities if the right-handed charged gauge boson is below the 100 TeV scale. In our realistic model, the linear seesaw can be verified at colliders if the related Higgs bidoublet is close to the TeV scale.