The Neutrinoless Double Beta Decay in the Colored Zee-Babu Model

We study the neutrinoless double beta decay in the colored Zee-Babu model. We consider three cases of the colored Zee-Babu model with a leptoquark and a diquark introduced. The neutrino masses are generated at two-loop level, and the constraints given by tree-level flavor violation processes and muon anomalous magnetic moment $(g-2)_{\mu}$ have been considered. In our numerical analysis, we find that the standard light neutrino exchange contribution can be canceled by new physics contribution under certain assumption and condition, leading to a hidden neutrinoless double beta decay. The condition can be examined comprehensively by future complementary searches with different isotopes.


I. INTRODUCTION
It is widely assumed that the tiny masses of neutrinos could be generated radiatively where neutrinos are Majorana particles. The Majorana neutrino mass models at two-loop level have been discussed in many previous works, e.g., Ref. [1][2][3][4][5][6][7][8], among which the Zee-Babu model [3,4] has attracted much attention. The addition of new particles in loops can bring us rich phenomena. However, whether neutrinos are Majorana particles or Dirac particles still remains unknown. The search for the neutrinoless double beta (0νββ) decay is the promising way to get us out of this dilemma.
The 0νββ decay can be realized if neutrinos are Majorana particles. If one only consider the standard light neutrino exchange, the inverse half-life has the form T 0νββ where G ν and M ν are the phase space factor (PSF) and nuclear matrix element (NME), m ee ≡ i U 2 ei m i is the effective neutrino mass, and m i (i = 1, 2, 3) are the masses of neutrinos. The most stringent limit on the 0νββ decay half-life in 136 Xe isotope is T 0νββ 1/2 > 1.07 × 10 26 yrs given by KamLAND-Zen experiment [9]. They obtained a constraint of | m ee | < 61 − 165 meV. The GERDA experiment has published their result T 0νββ 1/2 > 1.8 × 10 26 yrs with isotope 76 Ge leading to a similar bound | m ee | < 79 − 180 meV [10]. The future 0νββ decay experiments CUPID-1T [11] and LEGEND-1000 [12] using 100 Mo and 76 Ge isotopes can push the half-life to 10 27 − 10 28 yrs, leading to a sensitivity to | m ee | of around 15 meV.
In the effective field theory approach, the 0νββ decay can be described in terms of effective low-energy operators [13][14][15][16][17][18]. The contributions to 0νββ decay can be divided into long-range mechanisms [19][20][21][22] and short-range mechanisms [23,24]. The long-range mechanisms involve light neutrinos exchanged between two point-like vertices, which contain the standard light neutrino exchange. The short-range mechanisms involve the dim-9 effective interaction and are mediated by heavy particles. The decomposition of the short-range operators at tree-level has been completely listed in [25], and at one-loop level has been discussed in [26]. The short-range mechanisms at the LHC have been considered in [27]. An analysis of the standard light neutrino exchange and short-range mechanisms has been given in [28].
In this work, we study three cases of the colored Zee-Babu (cZB) model with a leptoquark and a diquark running in the loops. We consider the realization of tiny neutrino mass and contribution to 0νββ decay, with the constraints given by tree-level flavor violation processes and (g − 2) µ considered. The B physics anomalies and some other phenomena in the cZB model have been explored in [29][30][31][32][33][34][35][36]. The long-range contributions given by the leptoquarks have been considered extensively in the previous discussion [18,37,38]. We focus on the short-range impact on neutrinoless double beta decay in this model. The simultaneous introduction of the leptoquarks and diquarks in the cZB model can lead to short-range contribution, which can interfere with the standard light neutrino exchange contribution resulting in cancellation.
We organize our paper as follows. In Sec. II, we show the three cases in the cZB model and briefly review the constraints on them. In Sec. III, we discuss the 0νββ decay in each case, including short-range mechanisms and standard light neutrino exchange. Finally, we give our conclusion in Sec. IV.
The corresponding quantum numbers of the particles in these cases are summarized in Table I. In the fermion weak eigenbasis, the Yukawa interactions of these cases can be written as

SM Particles Quantum Number New Particles Quantum Number
(2) L symmetry, and αβγ is the Levi-Civita symbol. The Yukawa coupling matrices z 1SL , z 1ω , and z 3ω are symmetric, z 3S is antisymmetric, while the other coupling matrices are arbitrary [39].
To study the phenomenologies, we rewrite the Lagrangian in the mass eigenbasis as, Here we follow the basis transition U j [40], where V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. The scalar potential involving the leptoquark and the diquark fields contains cubic terms The two-loop diagrams which generate tiny neutrino masses in the colored Zee-Babu model. The left diagram corresponds to cases 1 and 2, and the right corresponds to case 3.
+2/3,β 2 For simplicity, we assume the quartic couplings of leptoquark and the SM Higgs doublet Φ to be vanishing for case 2 and case 3. In addition, the quartic couplings of ω 3 and Φ are also assumed to be negligible. The leptoquark/diquark multiplets are then degenerate in mass, we denote the masses of S 1 , S 2 , S 3 , ω 1 , and ω 3 as M S 1 , M S 2 , M S 3 , M ω 1 , and M ω 3 , respectively. In our numeraical analysis, the masses of the leptoquarks and diquarks are taken as M S 1.5 TeV and M ω 8 TeV, which accords with the bounds given by the ATLAS and CMS collaboration [41][42][43][44][45][46][47].

A. Neutrino masses
In the colored Zee-Babu model with a leptoquark and a diquark, the neutrino masses can be generated at two-loop level with down-type quarks running in the loop as shown in Fig. 1. The left Feynman diagram corresponds to case 1 and case 2, and the right one to case 3. The neutrino mass matrix elements in flavor basis take the form [29] where m D l is the mass of the l-th generation down-type quark. The superscripts k, l, m, n = 1, 2, 3 and the subscripts a, b, c = 1, 2, 3 of the couplings are neglected to keep the expression concise. The expression can apply to all three cases. Note that the coupling y S(L) equals y 1SL in case 1, y 3S in case 2, and y 2S in case 3. The I lm in Eq. (13) is loop integral where M S denotes leptoquark S i mass and M ω is diquark ω i mass in different cases. The integral can be simplified as [48] Feynman diagrams of neutron-antineutron oscillation (left) and proton decay p → π 0 e + (µ + ) (right) in case 1 and case 2. For case 1, with theĨ(r) can be calculated through the numerical integral way. The neutrino mass matrix can be diagonalized by the PMNS matrix U where m 1,2,3 are the masses of active neutrinos ν 1,2,3 .

B. Neutron-antineutron oscillation and proton decay
With the leptoquark coupling z 11 S = 0 (z S corresponds to z 1SR and V * z 1SL in case 1 and V * z 3S in case 2), case 1 and case 2 can lead to neutron-antineutron oscillation, as shown in Fig. 2. The transition rate of neutron-antineutron oscillation τ −1 is proportional to |(z 11 S ) 2 z 11 ω |. Using the current limit τ ≥ 4.7 × 10 8 s given by the Super-Kamiokande (Super-K) experiment [49], one can get the bounds With M S ∼ 1.5 TeV and M ω ∼ 8 TeV, |(z 11 S ) 2 z 11 ω | 10 −14 . The construction of operators and detailed calculation of neutron-antineutron oscillation can be found in [50][51][52][53][54][55][56][57][58]. However, the proton will decay when z 11 S and y 11 S are both set nonzero. For example, the non-zero couplings can contribute to the process p → π 0 e + (µ + ) as shown in Fig. 2. With the experimental limit τ /B(p → π 0 e + ) > 2.4 × 10 34 yrs and τ /B(p → π 0 µ + ) > 1.6 × 10 34 yrs given by Super-K [59], there is a strict bound on the couplings with the matrix element inputs from lattice [60,61] where y S denotes V * y 1SL and y 1SR in case 1 and V * y 3S in case 2. One can find that if M S ∼ 1.5 TeV and y 11 S ∼ 0.01 − 1, then z 11 S(L/R) needs to be at O(10 −26 − 10 −24 ) scale to avoid an inappropriate proton decay, leading to an unobservable neutron-antineutron oscillation. So in the discussion of cases 1 and 2, we assume z S = 0 to escape proton decay.

C. The texture setup and constraints
We show our texture zeros setup of the couplings matrices and list the bounds on the couplings, which are related to 0νββ decay in this subsection. The bounds are derived from muon anomalous magnetic moment and tree level flavor violation processes with four-fermion interactions considered.

Texture setup
The standard parameterization of the PMNS matrix is where c ij (s ij ) denotes cos θ ij (sin θ ij ), δ is the CP phase, and η 1,2 are the extra phases if neutrino are Majorana particles. The best fit values of these neutrino oscillation parameters have been derived in [62][63][64][65]. As there are no information about the Majorana phases ranges, they can varies from 0 to 2π freely. To evade constraints from various lepton flavor violation (LFV) processes, we adopt the Yukawa coupling matrices in case 1 as The matrix y 1SL and z 1ω are set to be complex and y 1SR is real. The contribution to neutrinoless double beta decay can survive when couplings (V * y 1SL ) 11 ≡ y 11 1SL , y 11 1SR and z 11 ω (in blue) are set nonzero. Moreover, we have set y 32 1SL , y 32 1SR = 0 (in red) to obtain the muon anomalous magnetic moment (g−2) µ . The entries of couplings y 23 1SL , y 33 1SL , z 33 1ω , z 23 1ω provide enough independent parameters to generate appropriate neutrino mass matrix. It is noted that the first component of the neutrino mass matrix is negligible under these entries with the constraint from the treelevel flavor violation processes |y 11 1SL | < 0.12, as shown in the following subsection, leading to an inconspicuous standard light neutrino exchange 0νββ decay. Hence we introduce y 31 1SL (in teal) to open the standard 0νββ decay.
The texture zeros setup of coupling matrices in case 2 and case 3 are similar to those in case 1 with V * y 3S ≡ y 3S . Though the coupling |y 11 2S | could be O(1) in case 3, we still choose the same form of the matrix to make the analysis consistently.

Constraints
The Yukawa couplings are constrained by tree-level flavor violation processes. It is natural to work with effective field theory where the effective Lagrangian can be described with four-fermion interaction operators as the new particles are at TeV scale. The effective Lagrangian involving S 1 leptoquark reads [31] The effective Lagrangians induced by S 3 andS 2 can be written as The constraints on the Wilson coefficient ijkn have been derived in [66][67][68][69], where with the constant equals 1 in case 1 and case 3 while the constant takes as 1 or 2 in the case 2. Here we have taken the bounds from [69] and have listed some of them in Table II which are related to the couplings that can contribute to neutrinoless double beta decay process. However, one needs to pay attention that bounds on the couplings |y ki S(L) y nj S(L) | can be derived from the numerical combined calculation of the bounds on |y ki S(L) y nj S(L) | and |y ki S(L) y nj S(L) | in case 1 and case 2. Moreover, the neutral meson mixing can be contributed by the diquarks. The B 0 d −B 0 d mixing needs to be concerned under our texture setup. The 95% allowed range of the couplings [70] is  , which can contribute to 0νββ decay. The bounds are taken from [69] with the unit of (M S /TeV) 2 .
The muon anomalous magnetic moments a µ = (g − 2) µ /2 can also give information on the couplings. The latest result of muon anomalous magnetic moment has been presented by Muon g − 2 collaboration [71] as which has a 4.2 σ discrepancy. The expression of muon anomalous magnetic moment in case 1 can be simplified as [72] ∆a While for case 2 and 3, the contributions are with the functions f i are defined in [72]. To explain the discrepancy, the couplings in case 1 have the relation Re[y 32 1SR y * 32 1SL ] ∼ 8.48 × 10 −2 with the leptoquark mass M S 1 = 1.5 TeV. However, in case 2 and case 3, the contribution to muon anomalous magnetic moment ∆a µ are negligible since there is no chiral-enhancement m q /m µ for these two cases.
After taking accounting of all the constraints mentioned, the parameter regions taken in our numerical analysis are and the z 23 iω (i = 1, 3) in each case are set to be |z 23 iω | < 1.5. We take µ = M S = 1.5 TeV and M ω = 8 TeV in our following discussion.

III. THE NEUTRINOLESS DOUBLE BETA DECAY
The 0νββ decay can be divided into short-range and long-range mechanisms. To study how short-range contributions impact the 0νββ decay in the cZB model, we briefly review the general formula of the short-range mechanisms via the effective field theory approach and give numerical analysis in this section.
A. The short-range 0νββ decay The short-range 0νββ decay operator can be written as O 0νββ ∝ uuddee, a dim-9 operator. The scalar mediated tree-level topologies and the decomposition of this operator has been listed in [25]. We follow the general parameterization of effective short-range Lagrangian in [23,28] where G F is the Fermi constant, m p is the proton mass, V ud is the ud component of the CKM matrix, and the dimensionless effective couplings are defined as χ i = XY Z i (X, Y, Z = R/L). The J and j, respectively, denote the quark and electron currents as One can express the effective operators in terms of the quark and electron currents as [24,25] The following expression gives 0νββ decay inverse half-life involving the short-range mechanism and light-neutrino exchange [28] T 0νββ where χ i are the effective couplings shown in Eq. (34), M XY i are the NMEs with the short-range mechanism and M ν is the NMEs with the light neutrino exchange. The dimensionless parameters    [73]. The PSFs numerical values of different isotopes are taken from [28], as shown in Table III. The first column is the lower limits for the decay half-life of different isotopes [9,10,[74][75][76][77]. The Table IV shows the values of light neutrino exchange NME M ν and short-range mechanism NMEs M XY i within the microscopic interacting boson model [28]. We just list the values of M XX i since both of the quark currents in the cZB models are right-handed, i.e., X = Y = R.
The effective operators in the three cases can be written as The corresponding Feynman diagrams in different cases are shown in Fig. 3. The effective couplings χ i in different cases are case 1 : RRL 3. The Feynman diagrams of neutrinoless double beta decay in the colored Zee-Babu model. The first row corresponds to case 1. The two diagrams in the second row coreespond to case 2 (left) and case 3 (right).
The effective neutrino mass can be related to the first component of the neutrino mass matrix [78] and takes form as The term related to m s cannot be neglected because we have assumed that there is a hierarchy among the couplings z ij ω .

B. Numerical results
Before we give our numerical results, it is necessary to notice that the QCD corrections can modify the NMEs [79][80][81][82][83][84]. If we consider the leading order QCD corrections and the numerical values of the RGE µ-evolution matrix elements with the same chiral quark currents [80] U XX the NMEs need to be recomposited as The inverse half-life [T 0νββ After considering the experimental values and substituting the numerical values of the PSFs and the NMEs shown in Table III and IV, one can get the limits on the couplings.
We show the contour with the effective neutrino mass | m ee | in the unit of eV and the couplings |z 11 ω (y 11 S ) 2 | in Fig. 4 and Fig. 5. The purple and green regions (upper and lower corners) are excluded by the 0νββ decay experiments KamLAND-Zen [9] and GERDA [10]. The red and blue regions correspond to the survival areas for the experiments CUPID-1T [11] and LEGEND-1000 [12] if we assume that no signals are found and set the half-lifetime to T 1/2 > 5.0 × 10 25 yrs (CUPID) and 5.0 × 10 26 yrs (LEGEND), whereas the inner darker areas relate to the sensitivities. We show all the three cases in Fig. 4, with the assumption |y 11 1SL | = |y 11 1SR | for case 1. Under this assumption, the term that contains RRL 1,2 and ν dominants in the inverse half-life expression. Due where the specific value of M ν /(β 1 − β 2 /4) varies from isotope to isotope. For the current experiments, the relation can be realized successfully due to the similar ratio value of M ν /(β 1 − β 2 /4) in 76 Ge and 136 Xe isotopes. Things will be different and intriguing when the next-generation experiments with 100 Mo isotope, e.g. AMoRE-II [85] and CUPID-1T, push the half-life to be order of 10 26 yrs. The slope of the band with 100 Mo is different from 76 Ge or 136 Xe, leading to the overlap of survival areas being narrowed. With the high sensitivity of the future CUPID-1T and LEGEND-1000 experiments, the survival band can be examined comprehensively. If there is no signal of 0νββ decays, the survival region will be reduced to the overlap area. On the other hand, if we see the signals in one experiment, the corresponding contour lines will be suitable. The other experiment can help us search for the appropriate region of the lines.
We show the contour of case 1 with assumption |y 11 1SL | |y 11 1SR | ≡ |y 11 S | in Fig. 5. This assumption is natural as the allowed regions have ten times difference, and the influence on the neutrino mass is negligible. The survival region is elliptical instead. The left panel refers to |y 11 1SR | = 10|y 11 1SL |. One can find that the constraint on the effective Majorana neutrino mass can be larger than the one with only standard neutrino exchange considered | m ee | 0.2 eV. The experiment with 100 Mo isotopes can help to reduce the survival area which is similar to what we discussed before. The right panel refers to |y 11 1SR | = 50|y 11 1SL | and the effect of the combined analysis with different experiments is not apparent. The bound on the couplings given by experiments (52) The limits will be more stringent in next generation 0νββ decay experiments, which have the potential to restrict |z 11 ω (y 11 S ) 2 | at O(10 −1 ) scale.

IV. SUMMARY
In this paper, we have discussed the neutrinoless double beta decay in the colored Zee-Babu model. We study all three cases for the colored Zee-Babu model with a leptoquark and a diquark. The tiny neutrino masses are generated at two-loop level, and neutrinoless double beta decay gets additional contribution from the leptoquarks. We set some texture zeros for the Yukawa coupling matrices to evade constraints from various lepton flavor violation processes. We obtain the allowed regions of parameters after considering the constraints given by tree-level flavor violation processed and charged lepton anomalous magnetic moment.
We have discussed the short-range and standard neutrino exchange mechanisms of neutrinoless double beta decay for each case. The short-range contribution can be realized at tree-level. The general formula of the short-range contributions via the effective field theory approach is briefly reviewed. We adopt the values of nuclear matrix elements calculated with the microscopic interacting boson model and consider the leading order QCD running correction. We give numerical analysis for the three cases with the current experimental results and sensitivities of next-generation experiments. We find that the neutrinoless double beta decay can be hidden with a linear relation in all the cases under certain conditions. The relation can be examined by future 0νββ decay experiments. The complementary analysis of the different isotope experiments can help reduce the overlap area of the survival region.