Decomposition of $d=9$ short-range $0\nu\beta\beta$ decay operators at one-loop level

We perform a systematical study of the dimension-9 short-range $0\nu\beta\beta$ decay operators at one-loop level. There are only six genuine topologies which generate eight diagrams, and the recipe to identify the possible one-loop realizations of the $0\nu\beta\beta$ decay operators is sketched. Certain hypercharge assignments are excluded by the absence of tree-level diagrams in a genuine one-loop model. The mediators of each decomposition can generate Majorana neutrino masses which are discussed up to two-loop level. We present an example of $0\nu\beta\beta$ decay model in which the neutrino masses are generated at two-loop level, and the short-range contribution can be comparable with the mass mechanism in some region of parameter space.


Introduction
The discovery of neutrino oscillations shows that neutrinos have tiny masses. In order to generate the tiny neutrino masses, one must extend the filed content or gauge symmetry of the standard model, or depart from renormalizability [1]. Depending on whether neutrinos are Dirac or Majorana particles, the neutrino mass terms can be constructed in different ways. For the case of Majorana neutrinos, neutrinos are identical with their antiparticles and the lepton number would be violated by two units ∆L = 2. As a consequence, the exchange of light Majorana neutrino can lead to the well-known neutrinoless double beta (0νββ) decay which violates the lepton number by two units as well. This is the standard mass mechanism. Conversely, if the rare 0νββ decay is observed, neutrinos must be Majorana particles [2].
The new physics giving rise to 0νββ decay will always produce a non-zero mass of Majorana neutrinos. Under the assumption that the Majorana neutrino mass is the unique source of 0νββ decay, the decay rate is fixed by the phase space factor, the nuclear matrix element and the effective Majorana mass m ββ with where U is the lepton mixing matrix and m i (i = 1, 2, 3) are the light neutrino masses. The effective mass m ββ is the (11) entry of the light neutrino mass matrix in the basis where the charged lepton mass matrix is diagonal. Hence 0νββ decay can provide valuable information on both neutrino mass spectrum and Majorana CP phases. The experimental search for the 0νββ decay is extremely challenging, and no positive signal has been observed so far. The most stringent bound is given by the KamLAND-Zen Collaboration with the half-life T 0ν 1/2 > 1.07 × 10 26 years for 136 Xe at 90% confidence level, this bound implies a limit on the effective Majorana neutrino mass |m ββ | < (61 − 165) meV [3]. The next generation tonne-scale 0νββ decay experiments such as LEGEND [4] and nEXO [5] aim at the half-life sensitivity of approximately 10 28 years. If all the future results from 0νββ decays, neutrino oscillation experiments and cosmology observation on the light neutrino mass sum are compatible with each other, it would be a strong support to the mass mechanism mentioned above.
However, it is notable that the 0νββ decay could possibly be dominantly induced by other new physics effects beyond that of Majorana neutrino masses. The contributions to the 0νββ decay rate can be generally divided into the short-range part [6] and long-range part [7]. See Refs. [8][9][10][11][12] for review. The short-range contributions arise from the lepton number violating couplings or the Majorana mass terms of new particles in possible extensions of the Standard Model (SM). The short-range dynamics of the 0νββ decay is expected to be at a scale Λ higher than the weak scale, and it can be described by a set of ∆L = 2 and baryon conserving operators which are invariant under the SM gauge symmetry SU (3) C × SU (2) L × U (1) Y . At leading order, there are 11 dimension-9 operators which give rise to 0νββ decay [13][14][15]: where Q = (u L , d L ) T and = (ν L , l L ) T stand for the quark and lepton doublets of SU (2) L respectively, and we denote the charge conjugation fields e c R = (e R ) c and c = ((ν L ) c , (l L ) c ) T . The Roman letters i, j, m, n = 1, 2 refer to SU (2) L indices, and is the 2-component antisymmetric tensor with 12 = − 21 = 1, and λ A are the Gell-Mann matrices.
Here we only consider operators involving first generation fields, since we are concerned with the operators which directly contribute to the 0νββ decay. In addition to 0νββ decays, these operators in Eq. (3) can produce the signals of same sign dilepton, single lepton + MET or 2jets+MET at hadron colliders [13]. The different possible ways to generate the above dimension nine (d = 9) 0νββ decay operators at tree level have been studied in Ref. [16]. All the tree-level decomposition of the long-range 0νββ decay operators has been discussed in Ref. [17]. The one-loop contribution to the 0νββ amplitude was studied in the radiative neutrino mass models [18][19][20][21]. In this paper, we shall study the systematical decomposition of the eleven operators O 0ν 1,2,...,11 at one-loop level by using the diagrammatic approach [22,23]. In this work, we consider the cases that the mediators are either scalar or fermion fields. The topologies of the diagrams for scalar and vector are the same, nevertheless vector fields are expected to be the gauge bosons of some new symmetry and their masses are generated through the spontaneous breaking of that symmetry. That is to say the SM gauge group should be extended to accommodate the new vector bosons, consequently we only discuss scalar messengers here. Furthermore, we assume the new fermions are vector-like so that the resulting models are anomaly free. This paper is organized as follows. In section 2, the general procedures of decomposing a non-renormalizable operator are sketched, and we discuss how to generate the topologies, diagrams and models of the short-range 0νββ operators. In section 3, the relation between 0νββ models and neutrino masses is discussed. In section 4, we give an example of one-loop model for short-range 0νββ decay, the leading contribution to neutrino masses arises at two-loop level, and the prediction for half-life is analyzed. Section 5 summarizes our main results. The complete set of 0νββ decay operators below the electroweak scale as well as the general formula for half-life are given in Appendix A. We list the non-renormalizable topologies and finite non-genuine diagrams in Appendix B. We present the possible one-loop models and the quantum number assignments of mediators for the 0νββ decay operator N4 with the diagram N-1-1-1 in Appendix C.

Generate genuine topologies, diagrams and models
The diagrammatic decomposition approach [22,23] allows us to systematically identify the possible ultraviolet (UV) completion of any effective high dimensional operator O EF in the SM effective theory, and it has been used to systematically classify the Majorana neutrino mass models generated at one-loop level [24], two-loop level [25] and three-loop level [26]. The classification of Dirac neutrino mass models have also been performed in this approach [27][28][29][30]. The general procedure of decomposing a non-renormalizable operator O EF at L− loop level is as follows: x Generate the L−loop connected topology with three-point and fourpoint vertices, and the number of the external lines should be exactly equal to the number of the fields contained in the considered operators. The topologies with tadpoles and selfenergies are generally discarded. y Assign the fields involved in the concerned operators to the external lines, and specify the Lorentz nature (spinor or scalar) of each internal line. Thus the topologies are promoted to diagrams. For the four-point vertex, the corresponding interaction is renormalizable if and only if all the four relevant legs are scalars, while the trilinear vertices can be the fermion-fermion-scalar and scalar-scalar-scalar interactions. We discard all the diagrams with non-renormalizable interactions. z Construct the UV complete models by specifying the quantum numbers of the internal fields, and each vertex should be invariant under the SM gauge symmetry.
Although the resulting UV models obtained in this way can generate the concerned effective operator O EF at L−loop level, one should guarantee that it is the leading order contribution. It is called a genuine model if there is no lower order contribution to O EF otherwise the model is non-genuine. The topologies associated with the genuine models is called genuine topologies. If all the UV models generated from a topology are non-genuine, the corresponding topology is non-genuine. In a similar fashion, one can define genuine diagrams and non-genuine diagrams. A diagram is non-genuine if it is possible to compress one or more loops into a renormalizable vertex [26], in this case, lower order contributions will be allowed. In general, any internal loop (or loops) with three external legs can be compressed into a three point vertex regardless of the field content of a model, thus the associated topology is non-genuine. It is only possible to compress a internal loop (or loops) with four external legs into a renormalizable vertex if all external lines are scalars. In the present work, we shall apply the above procedures to study the one-loop decomposition of the neutrinoless double decay operators in Eq. (3).

Generate topologies
All the dimension 9 effective 0νββ decay operators involve six quark and lepton fields. We first use our own code to generate the connected one-loop topologies with six external legs. All the topologies with tadpoles and self-energy are excluded, because their loop integrals are divergent and thus lower order counter term is required in a consistent renormalization scheme. After excluding the topologies which contain an internal loop with three legs, we find there are totally 43 topologies shown in figure 1 and figure 14. Only the six one-loop topologies in figure 1 can lead to genuine diagrams (models). The topologies in figure 14 need non-renormalizable interactions to accommodate six external fermion lines. For the sake of completeness, we show the non-genuine topologies which contain an internal loop with three legs in figure 15. We see that all these non-genuine topologies can be regarded as extensions of the tree-level 0νββ decay models, where one of the vertices is generated at one loop. As regards the tree-level decomposition of the 0νββ decay operators, there are only two renormalizable topologies [16] which are displayed in figure 1. Generally three mediators are needed in the tree-level UV complete models and their quantum numbers are unambiguously fixed. We also provide the tree-level decomposition in the present work, since these results are essential when determining the genuineness of a one-loop 0νββ decay model.
Figure 1: Genuine topologies which can lead to tree-level and one-loop genuine models for the short-range part of the 0νββ decay amplitude.

Generate diagrams
All the six external legs for 0νββ decay are quark and lepton fields. We proceed to insert fermion or scalar into each internal line, and Lorentz invariance requires that each vertex must contain even number of fermion. The UV complete models are assumed to be renormalizable, thus the interaction vertices should be dimension-four three-point and four-point vertices, and consequently the interaction terms in the Lagrangian are of the form where F i and S i denote a generic fermion and scalar fields respectively. A topology is non-renormalizable if the diagrams for all possible fermionic and scalar insertions contain non-renormalizable vertex. We show the non-renormalizable topologies in figure 14 for completeness. Some diagrams arising from a topology are possibly superfluous. One needs to consider all possible permutations of fermion and scalar lines and compare the couplings of the vertices. If the couplings of two diagrams match with each other, they are exactly the same diagrams. For the genuine topologies in figure 1, we show the corresponding independent diagrams in figure 2. Both topologies N-1-4 and N-1-5 give rise to two diagrams, while there is only one diagram for each of the remaining four topologies N-1-1, N-1-2, N-1-3 and N-1-6. Because all the external legs of the short-range 0νββ decay are fermions, the four-point interaction S 1 S 2 S 3 S 4 is only involved in the diagram N-1-1-1 and the interaction vertices are of the forms F 1 F 2 S 1 and S 1 S 2 S 3 in other cases, as can be seen from figure 2.
In particular, it is impossible to compress the loop into a renormalizable vertex, hence we expect all these diagrams in figure 2 could give genuine models. We also show the tree-level diagrams which lead to 0νββ decay in figure 2, since they are essential when determining whether a loop model is genuine. There are only two diagrams at tree level, and the Lorentz natures of all mediators are completely fixed.

Generate models
The next step is to generate models for the diagrams displayed in figure 2, and we need to specify the SU (3) C , SU (2) L and U (1) Y quantum numbers to each line in these diagrams. Firstly we attach the fields in the 0νββ decay operators to external lines of the diagram, subsequently gauge invariance allows us to determine the quantum numbers of each internal line. Notice that there are only finite possibilities for the quantum numbers of the internal fermions and scalars in the tree-level models. However, any one-loop diagram in figure 2 leads to an infinite number of genuine models. For illustration, we shall impose an upper limit on the dimensions of the SU (3) C and SU (2) L representations assigned to the mediator fields. The details of how to generate models are given in the following subsections.

Attach external fields
The eleven 0νββ decay operators in Eq. (3) can be divided into six categories as listed in table 1, and the operators of each category involve the same fields as well as the same UV completion. The vector fields should be gauge bosons of certain gauge symmetry and their masses are generated by the spontaneous breaking of symmetry. Thus it would be necessary to extend the SM gauge group if vectors are mediators. Therefore we focus on the scalar and fermion mediated cases in the present work, the cases of vectors as mediators would be not considered. The SM lepton and quark fields are chiral fields in weak basis, Lorentz invariance Notation 0νββ decay operators External fields requires vector mediators for certain attachment of external fields, and these cases would be dropped. As an example, we shall consider the one-loop UV completion of the 0νββ decay operator N4 based on the topology N-1-1. Let us consider the diagram in figure 3, there are six possible ways to attach the fields of the operator N4 to the external lines of the vertex A. If the two external legs are assigned to be u R , d R or e R , d R , the non-vanishing fermion bilinears areū R γ µ d R andē R γ µ d R respectively, consequently the internal leg has to be a vector field X µ to form a Lorentz invariant interactionū R γ µ d R X µ andē R γ µ d R X µ . As a result, these two cases will be discarded. Since we can freely attach the fields to the external lines, therefore there are generally multiple external line structures for a given diagram and some of them are superfluous. The redundant assignments of external lines must base on the same diagram. In order to find out the redundant ones, we need to consider the possible permutations of vertices and compare the interlinkage between vertices. If several kinds of attachments have the same structure of interlinkage, they will be identified as one.

Determine U (1) Y charge
Once the external legs are specified, the hypercharges fulfilling of the internal lines can be determined by imposing the hypercharge conservation vertex by vertex. Note that the hypercharge of a field is related to its electric charge via the Gell-Mann-Nishijima formula Q = T 3 +Y , where T 3 is the third component of the weak isospin. For example, the equations of hypercharge conservation for the diagram N4-1-1-1-1 read as with The solution to the above equations is given by where α is arbitrary real parameter. A definite value of α should be taken in a concrete model. In a similar way, we can determine the hypercharges for the other diagrams in figure 4 as follows, Figure 3: The possible ways of attaching the fields of the operator N4 to the external legs of the vertex A in diagram N-1-1-1, where the dashed lines and wavy lines on the right side denote scalar and vector fields respectively. Notice that Lorentz invariance fixes the mediator to be scalar or vector because the external legs are fermions with definite chirality. Figure 4: The possible ways to attach the fields of the operator N4 to the diagram N-1-1-1.
We see that a hypercharge flow encoded in the parameter α in the loop is allowed.

Assign SU (2) L transformation
We proceed to assign the SU (2) L transformations to the internal lines, and they are denoted by the dimensions of the SU (2) L representations. For the concerned renormalizable interactions, the invariance under SU (2) L gauge symmetry requires the following conditions should be fulfilled, where n F i and n S i denote the SU (2) L representations to which the fields F i and S i are assigned. Notice that each SU (2) representation and its complex conjugate are equivalent. Moreover, it is necessary to check that some 0νββ decay operators can really be generated after integrating out the mediators. For example, let us consider the models in figure 5, the external legs are Q, Q,ū R ,ū R , , c which coincide with those of the operators O 0ν 3 and O 0ν 4 . In the left panel, the renormalizable interaction¯ c S at the vertex A is vanishing because the SU (2) singlet contraction of two doublets is antisymmetric. The model in the right panel of figure 5 leads to the effective operatorē L ν c LūR u LūR d L which can not induce the 0νββ decay process, although its interaction vertices are invariant under SM gauge symmetry. As a consequence, we need to explicitly expand the SU (2) L index of the interaction at each vertex, contract the internal lines and see whether the 0νββ decay operator can be produced. For illustration, we shall consider the SU (2) L singlet, doublet and triplet assignments for the messenger fields in the present work, and higher dimensional representations can be discussed in the same fashion.
The SM quantum number assignments for the internal fields are independent of their Lorentz property, therefore this step can be performed at the topology level. In table 2, we list the possible SU (2) L assignments of fields for the topology N-1-1. For the set of operators N1, N2 and N3, four of the external fields are SU (2) L doublets and the other two are singlets, consequently there are four different assignments of external legs, as shown in table 2. The external lines of the operator N4 are all singlets under SU (2) L . The fields of the operators N5 and N6 include two doublets and four singlets of SU (2), and there are four different assignments of external fields as well. The possible SU (2) L quantum numbers of the internal fields can be fixed by the invariance condition in Eq. (7), and it is necessary to check that the 0νββ decay operators can really be produced, as explained above.  Figure 6: Field assignments and color flow for the topology N-1-1, where E i and I i denote the external and internal fields respectively.  Table 2: Assign the SU (2) L quantum numbers to fields of the topology N-1-1 in figure 1.
The labels E i and I i of the external and internal lines are showed in figure 6.

Assign SU (3) C transformation
Then we discuss the possible transformation properties of the internal fields under SU (3) C , we consider the SU (3) representation up to dimension 8, namely, 1, 3,3, 6,6 and 8, high dimensional representations can be discussed analogously. All the 0νββ decay operators in Eq. (3) involve two quarks, two anti-quarks and two lepton fields which are in the irreducible representations 3,3 and 1 of the SU (3) C group respectively. We assign SU (3) C quantum numbers to each internal line, then consider whether SU (3) C invariant contractions can be formed at the vertices by using the tensor product reduction rules such as We shall use the Mathematica package GroupMath [31] to determine the SU (3) C gauge invariance of the interaction vertices. The renormalizable interaction compatible with SU (3) C gauge symmetry could be vanishing, in particular, when the repeated fields are involved. We take the interaction displayed in figure 7 as an example, which is of the form d c R d R S, with S being a color triplet. Expanding the color index of the down quark explicitly, we obtain where r, g, b denote the three different colors of quark fields. Hence it is necessary to explicitly expand the SU (3) C indices of each interaction vertex to see whether point interactions are vanishing. The SU (3) C quantum numbers of the internal fields can be discussed at the topology level as well. We list the possible color charge assignments of the topology N-1-1 in table 3, where the SU (3) C representations up to dimension 6 are considered due to limited space and the results including octet assignment are given in the attachment [32]. The labels of the fields and the color flow of the topology N-1-1 are shown in figure 6.
As explained above, we can start from a diagram in figure 2, attach the fields of the 0νββ decay operators to the external legs, subsequently determine the possible U (1) Y , SU (2) L and SU (3) C quantum numbers of the internal fields by imposing gauge invariance at each vertex, then the one-loop UV complete models for the effective 0νββ decay operators can be obtained. The different models are named as Ni-a-b-c-d-e-f, where Ni with i=1, 2, 3, 4, 5, 6 denotes the set of 0νββ decay operators, the second index "a" refer to tree (a=0) and oneloop (a=1) models, the symbols "b" and "c" stand for the topology and diagram respectively, the index "d" indicates the attachment of external fields, and the last two indices "e" and "f" denote the assignments of the SU (2) L and SU (3) C respectively. For instance, combining figure 4 with the SM electroweak-sector quantum number assignments in table 2 and table 3, we can obtain all the models arising from the diagram N-1-1-1. The possible UV completions for the operator N4 are listed in table 6, table 7, table 8, table 9 and table 10.

Determine genuineness
Numerous one-loop models for short-range 0νββ decays can be generated through the algorithm described in previous sections, in particular all the tree-level 0νββ decay models are generated. A one-loop model is the dominant contribution if and only if the new messenger fields are insufficient to mediate the more important tree-level diagram, then such model would be called genuine. In other words, the fields which generate the tree-level 0νββ decays should be absent in a genuine one-loop model. If the lower order contributions can not be forbidden without extra symmetry, the model would be non-genuine. We determine the genuineness of each model by comparing its field content with that of tree-level models one by one. Since the quantum numbers of the internal particles in the tree-level 0νββ decay models are unambiguously fixed, genuineness of a one-loop model generally excludes certain value of the hypercharge parameter α. We take the model N1-1-1-1-2-1-1 for illustration, see figure 8, we have defined the notation C Y L for the quantum numbers in the diagram, where C refers to the SU (3) C transformation, L refers to the SU (2) L transformation, and Y stands for U (1) Y charge. When the hypercharge parameter α = ±1, the mediators S 2 and S 3 and the associated interactions allow to generate the tree-level 0νββ decay models N1-0-1-1-2-1-1, N2-0-1-1-1-1-1, N3-0-1-1-3-1-1 which are the more important ones, as shown in figure 8. Therefore the genuineness of the one-loop model N1-1-1-1-2-1-1 requires α = ±1. The condition of genuineness has been considered for each possible one-loop decomposition of the 0νββ decay operators, see the attachment for the full results [32].

Neutrino masses in 0νββ decay models
We proceed to discuss the neutrino masses which are generated in the above models of shortrange 0νββ decay. The famous black box theorem establishes that the models which can lead to 0νββ decay must produce Majorana neutrino masses at or below four-loop order [2]. The four-loop diagram guaranteed by the black box theorem produces a very tiny neutrino mass of order m ν ∼ 10 −24 eV which is too small to explain the current neutrino experimental Figure 8: One example of a non-genuine model N1-1-1-1-2-1-1, the mediators S 2 and S 3 will also lead to tree-level contributions N1-0-1-1-2-1-1, N2-0-1-1-1-1-1, N3-0-1-1-3-1-1 when the hypercharge parameter α = ±1. The quantum numbers are given in the notation C Y L , where C refers to the SU (3) C transformation, L refers to the SU (2) L transformation, and Y stands for U (1) Y charge. data [33]. However, generally the neutrino masses are generated at a lower loop order in a specific 0νββ decay model so that the correct order of neutrino masses could be obtained. It is well known that the exchange of a light neutrino between two SM charged current interactions could lead to 0νββ decay, and it is usually called mass mechanism which is characterized by the effective neutrino mass m ββ in Eq. (2). For models with tree and oneloop neutrino masses, the mass mechanism is generally expected to dominate unless the model parameters are fine-tuned. The short-range contribution could be comparable with the mass mechanism for models with two-loop and three-loop neutrino masses [34]. One can construct neutrino mass diagrams by using the SM fields and the mediators which appear in one-loop renormalizable 0νββ decay models. In order to identify the neutrino mass diagrams, one need to compare the messenger fields of the 0νββ decay models with those of neutrino mass models. In the present work, we are concerned with the neutrino mass models up to two-loop level which can be generated from 0νββ decay models. Figure 9: The neutrino mass models generated from the 0νββ decay model N1-1-1-1-1-1-1. The one-loop neutrino mass diagram W-1-2-2-1-1-1 can be generated by the mediators of N1-1-1-1-1-1-1 when the hypercharge α = −7/6, 5/6, and the two-loop neutrino mass diagram W-2-1-2-1-11-7 can always be generated.

An example model of one-loop 0νββ decay
In this section, we present an example of model N1-1-6-1-4-1-1 which is a decomposition of the 0νββ decay operator N1 based on the diagram N-1-6-1 in figure 2. We add six new fields to the SM particle content in this model, and their quantum number assignments are listed in table 4. The interaction Lagrangian for these fields should respect the SM gauge symmetry, and it is of the following form where τ 2 is the Pauli matrix, and iτ 2 = is the 2-component antisymmetric tensor. This Lagrangian can generate the one-loop diagram for 0νββ decay shown in figure 10. The gauge invariant scalar potential terms involving new scalars are given by We denote S 3 = (S 3,1 , S 3,2 ) T , the first term in Eq. (17) can induce mixing between the two states S 1 and S 3,1 after electroweak symmetry breaking. The bilinear terms of S 1 and S 3,1 read as where v H is the vacuum expectation value of the Higgs field with H 0 = v H , we have redefined m 2 S 1 and m 2 S 3 to absorb the contributions of λ S 1 H v 2 H and λ S 3 H v 2 H respectively. Assuming the mixing parameter µ to be real, the above mass matrix can be diagonalized by a two-dimensional rotation matrix where Accordingly the mass eigenstates are related to the interaction basis by Rotating the interaction in Eq. (16) to the scalar mass eigenstate basis, the terms relevant to 0νββ decay are given by The one-loop diagram in figure 10 can be straightforwardly calculated, and we find the following effective operator is generated where J R,L =ū(1 ± γ 5 )d, j L =ē(1 + γ 5 )e c are defined in Eq. (38), and the coefficient LRL 1 is given by The function F(x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) is the loop integral and it is given by In the limit (25) is not valid anymore. In this case, we can use the following formula instead,  [36] if only the short-range mechanism is considered.

Prediction for neutrino masses
We have checked that for this model the leading order neutrino masses arise from a two-loop model W-2-5-4-1-3-10, as shown in figure 10. The relevant interaction Lagrangian is given by The lepton number is violated by the simultaneous presence of the above terms. After the electroweak symmetry breaking, the above interaction can be written in the mass eigenstate basis as follows, where the unitary rotations of the down type quarks to mass eigenstates have been absorbed into the couplings y Iα and y V α . The neutrino mass matrix arises from the two-loop diagrams of figure 11 in the mass basis and it can be straightforwardly calculated as Here the loop integral I r a , t α , r b , t β , r c is (30) with The momentum space integration T 3 (x 1 , x 2 , x 3 ) can be performed through the introduction of Feynman parameters, and it has been calculated in Refs. [37][38][39][40]. The analytical expression of I r a , t α , r b , t β , r c reads as where with Li 2 (x) being the dilogarithm function ln(1 − y) y dy (34) and The neutrino mass in Eq. (29) is dominated by the contribution of the bottom quark because the hierarchical mass pattern m b m s m d . Only one generation for every new fermions and scalars are introduced in our model, therefore the neutrino mass matrix of Eq. (29) is of rank two so that the lightest neutrino is massless. If one allows for more than one generation of the new fermions and scalars, the corresponding neutrino mass matrix can be read out analogously and the mass of the lightest neutrino could be non-zero. Figure 11: The diagrams for the neutrino mass generation in the model W-2-5-4-1-3-10, which is mediated by the fields of the 0νββ decay model N1-1-6-1-4-1-1, where d k with k = 1, 2, 3 stand for the down quark, strange quark and bottom quark respectively.

Half-life time of 0νββ decay
As shown in previous subsection, the mediators of the 0νββ decay model can generate nonvanishing light neutrino masses at two-loop level. Therefore both the short-range operator in Eq. (23) and the mass mechanism of the light neutrino exchange contribute to the 0νββ decay, and the two contributions should be added coherently. Using the general formula of Eq. (40) for the inverse half-life, we have where α and β denote the complex phases of m ββ and LRL 1 respectively, note that the nuclear matrix elements are conventionally defined to be real. We see that the interference between the short-range contribution and mass mechanism is constructive for α − β = 0 and destructive for α − β = π. It is remarkable that the two contributions would exactly cancel each other if LRL 1 = −m ββ M ν /(m e M LR 1 ). We show the constraints on the effective Majorana neutrino mass m ββ and the shortrange coupling | LRL 1 | in figure 12, where the values of phase space factor and nuclear matrix elements are adopted from Ref. [36]. The highlighted areas denote the allowed regions by the current limits and future sensitivities of the 0νββ decay half-life of the isotopes 76 Ge and 136 Xe, where the phase α − β freely varies between 0 to π. The half-life sensitivity would be increased by a factor of approximately one hundred in upcoming generation of tonne-scale experiments, therefore the constraint on parameter space would be improved considerably, as can be seen from figure 12. Moreover, we notice that the constraint imposed by 136 Xe is a bit more stringent than that of 76 Ge. Cosmology has played an important role in constraining neutrino properties, and the analysis of cosmological observables can provide an upper bound on a total neutrino mass. Within the standard minimal ΛCDM model, the present cosmological bound on neutrino mass is i m ν i < 0.12 eV for Planck TT, TE, EE+lowE+lensing+BAO [41]. Taking into account the values of the neutrino mass squared differences and mixing angles measured by the neutrino oscillation experiments [42], one can obtain that the effective Majorana neutrino mass is in the region: |m ββ | ≤ 31.12 meV for normal ordering (NO) and 18.62 meV ≤ |m ββ | ≤ 51.14 meV for inverted ordering (IO). With this constraint the current bounds T 1/2 ( 76 Ge) > 1.8×10 26 yr [35] and T 1/2 ( 136 Xe) > 1.07×10 26 yr [3] require the coupling | LRL 1 | smaller than 4×10 −10 . It is remarkable that the forthcoming tonne-scale searches for 0νββ decays would constrain the parameter space into a narrow band where the short-range mechanism is comparable with the mass mechanism.
We proceed to discuss the relative size of the short-range mechanism and mass mechanism to the decay rate. We take m S 1 = m S 2 = m S 3 = m F 1 = m F 2 = m F 3 = 1 TeV and the coupling constants y Ik = y III1 = y IIe = y IV 1 = y V k = y V Ie ≡ y ef f to be real. We display the contour region of log 10  figure 13, where ν ≡ m ββ /m e . The parameter µ dictates the mass splitting of m 2 S n1 and m 2 S n2 , as shown in Eq. (20). If µ is quite small with µ/TeV 1, m 2 S n1 and m 2 S n2 tend to be degenerate so that cancellation between two terms of the light neutrino mass matrix in Eq. (29) occurs. As a result, the mass mechanism is generally sub-dominant to the short-range contribution for µ/TeV 1. Furthermore, from Eq. (24) and Eq. (29) we see LRL Therefore the mass mechanism is dominant in the region y ef f 1 while the short-range mechanism is dominant in the region y ef f > 1.

Conclusion
The 0νββ decay is the most sensitive experiments to establish that the massive neutrinos are Majorana or Dirac particles. If the signal of 0νββ decay is observed in the future, lepton number would be violated by two units and neutrinos must be Majorana particles. The rare 0νββ decay process might also provide important information on the neutrino mass spectrum, the Majorana phases and the absolute scale of neutrino masses. On the other hand, if neutrinos are Majorana particles, 0νββ decay can be mediated by exchange of a  [5]. The gray bands denote the regions excluded by the cosmological data 3 i=1 m ν i < 0.12 eV at 95% confidence level from the Planck Collaboration [41], and the oscillation parameters are fixed to the current best fit values [42].  Figure 13: The relative size of the short-range mechanism and mass mechanism to the decay rate in the plane µ versus the couplings y ef f . The mass mechanism is dominant in the blue region while the short-range mechanism is dominant in the dark red region. The shaded regions denote the contours of the current limit and future sensitivities in 136 Xe (left) and 76 Ge (right) 0νββ decay search, where the phase α − β freely varies between 0 and π.
light neutrino between two charged current vertices, and it is the so-called mass mechanism. Additional contributions from short-range mechanism is also possible, where the exchanged particles are much heavier than the 0νββ decay energy scale typically 100 MeV, and they can be represented by non-renormalizable contact interactions with six external fermions.
In the present work, we have systematically studied the possible one-loop UV completions of dimension-9 short-range 0νββ decay operators in Eq. (3). The general procedure to be followed involve three steps: firstly generate one-loop topologies with six legs, then promote topology to diagram by specifying the Lorentz nature of fields, finally attach fields to external lines and determine the SM quantum numbers of internal particles by gauge invariance. Notice that there are infinite possibilities for the quantum numbers of the internal fermions and scalars, thus a given diagram can lead to an infinite number of UV completions. To be more specific, excluding the topologies with tadpole and self-energy, there are only 48 one-loop connected topologies with three-point and four-point vertices and six external lines, as shown in figure 1, figure 14 and figure 15. Only 11 of them in figure 1 and figure 15 can accommodate 6 external fermion lines using renormalizable interactions. The 5 topologies in figure 15 contain a one-loop sub-diagram which can be compressed into a three-point vertex so that the more important tree-level contributions can not be forbidden. As a result, it is sufficient to only consider the 6 one-loop topologies in figure 1 which lead to 8 one-loop genuine diagrams shown in figure 2. One can determine the transformations of the internal fields under SM gauge group by imposing gauge invariance at each vertex, the SU (3) C , SU (2) L representations and hypercharge U (1) Y assignments can be unambiguously fixed in tree-level realizations while there are an infinite number of UV completions at one-loop level. The complete list of the models together with the internal fields and their quantum numbers are given in the supplementary file [32], where the SU (2) L and SU (3) C representations with dimensions less than 4 and 10 respectively are considered.
In order to guarantee the dominance of the one-loop contribution to 0νββ decay, the mediators of the tree diagrams should be absent in the one-loop models. As a consequence, certain values of the hypercharge parameter α are excluded. The 0νββ decay models violate lepton number, thus they can also generate Majorana neutrino masses, as is required by the black box theorem [2]. Therefore both short-range contribution and mass mechanism are present in these models. We have discussed the Majorana neutrino masses up to two-loop level which can be generated from the mediators of the 0νββ decay models. Furthermore, we give a concrete one-loop 0νββ decay model in which the neutrino masses are generated at 2-loop level. The predictions for the neutrino mass matrix and the half-life of 0νββ decay are presented. The short-range contribution can be as important as the mass mechanism in some region of parameter space.
There are many experiments that are searching for 0νββ decay or are in various stages of planning and construction over the world. We expect it would become an important probe for physics beyond the SM. If the observation of 0νββ decay experiments is in conflict with neutrino mass ordering measured by neutrino oscillation experiments and the neutrino mass bounds from β decay and cosmology in the future, the new contributions beyond the mass mechanism would be necessary. Then our results can be used to construct short-range 0νββ decay models. Furthermore, generally a few messenger fields are required in the one-loop 0νββ decay models, it is interesting to study the phenomenological implications of these new fields in collider and cosmology, and this is left for future work.

A 0νββ decay operators below the electroweak scale and half-life
The SM gauge symmetry SU (3) C ×SU (2) L ×U (1) Y is spontaneous broken down to SU (3) C × U (1) EM . The leading operators which contribute to 0νββ decay at short distance involve four quark fields and two charged leptons, they respect SU (3) C × U (1) EM symmetry and can be written as the product of three fermion currents [6,36] ( Here the normalization factor G 2 F cos 2 θ C 2m P has been omitted, where m P is the proton mass and θ C is the Cabibbo angle. The factor cos 2 θ C is included so that the phase space factor of short-range contribution can be defined in the same way as that for the standard light neutrino exchange [36]. The hadronic current J and leptonic current j are bilinears of quarks and electrons, where a = 1, 2, 3 is SU (3) C color indices and σ µν = i 2 [γ µ , γ ν ]. The chirality assignment in the charged lepton current j R,L is flipped, and the subscripts L and R are associated with 1 + γ 5 and 1 − γ 5 respectively. Thus the operators j R and j L describes the creation of two right-handed and left-handed electrons respectively. Notice that using the Fierz identity one can show that both operators (O 2 ) {LR}L = J µν L J Rµν j L and (O 2 ) {LR}R = J µν L J Rµν j R are vanishing. Hence the most general short-range interaction Lagrangian responsible for 0νββ decay can be written as where the indices χ 1 , χ 2 , χ are L or R and they denote the chiralities of the quark and electron currents in the order of appearance in Eq. (37). Under the presence of both standard mass mechanism and the short-range mechanism, the inverse 0νββ decay half-life time can be expressed as [36] T −1

B Non-renormalizable topologies and finite non-genuine diagrams
For completeness, we list the topologies which always lead to the non-renormalizable diagrams in figure 14 as well as the non-genuine topologies in figure 15. The non-genuine model arising from the topologies in figure 15 can be regarded as extensions of the tree-level 0νββ decay models, and one of the vertices is generated at one-loop level, as illustrated in figure 16 and figure 17.

C Possible UV completions for the 0νββ decay operator N4 based on the diagram N-1-1-1
In this appendix, we present the possible one-loop models which generate the 0νββ decay operator N4, and the models are based on the diagram N-1-1-1. The SM quantum numbers of the messenger fields are listed in tables 6-10.  Table 5: Matching between the 0νββ decay operators above the electroweak scale and those below the electroweak scale. Figure 15: The one-loop topologies leading to non-genuine diagrams. Figure 16: Non-genuine one-loop diagrams which correspond to the one-loop generation of one of the couplings in the tree-level diagram N-0-1-1(see figure 2). The diagrams in the box are divergent. Figure 17: Non-genuine one-loop diagrams which correspond to the one-loop generation of one of the couplings in the tree-level diagram N-0-2-1 (see figure 2). The diagrams in the box are divergent.

Model Name
Model Name  Table 8: The one-loop models for the 0νββ decay operator N4 arising from the diagram N4-1-1-1-3 shown in figure 4.
Model Name