$\Delta B=2$ neutron decay into antiproton mode $n\to \bar pe^+\nu(\bar\nu)$

We discuss the unique baryon number violation by two units neutron decay mode $n\to \bar p e^+\chi$, with $\chi$ being the standard model (SM) neutrino $\nu$ or antineutrino $\bar\nu$ or any beyond SM light fermion, in the framework of effective field theory. This mode is kinematically allowed but rarely discussed theoretically or searched for experimentally. We estimate the lower bound on its partial lifetime from that of the dinucleon decay $np\to e^+\chi$ per oxygen nucleus $^{16}$O set by the Super-Kamiokande experiment, with a conservative bound $\Gamma^{-1}_{n\to\bar pe^+\chi}>5.7\times 10^{39}~\rm yrs$. We also discuss its characteristic signature for the future experimental search and astrophysical implications.

Introduction. The baryon-antibaryon asymmetry of the universe requires the violation of baryon number (B), which is one of the three Sakharov conditions for a successful baryongenesis mechanism [1]. The baryon number violation (BNV) is also a general feature in scenarios of physics beyond the standard model like the grand unified theories [2][3][4], in which the |∆B| = 1 single nucleon decays like proton decay p → e + π 0 are predicted. However, the scale of new physics (NP) associating with |∆B| = 1 nucleon decays is constrained, by the current experimental data, to be around 10 14−16 GeV, which is impossible to directly produce those heavy particles at current and future high energy colliders.
On the other hand, there is a class of scenarios suppressing |∆B| = 1 nucleon decays but contribute dominantly to |∆B| = 2 processes like the neutron -antineutron (n−n) oscillation, hydrogen -antihydrogen (H−H) oscillation and dinucleon to dimeson/dilepton decays in nuclei (like pp → π + π + , e + e + ) [5][6][7][8][9]. In such scenarios, the associated NP scale is lowered to be around O(1 − 10 4 ) TeV from the current experimental results on n−n oscillation [10,11] and dinucleon to dimeon/dilepton decays [12][13][14], and this TeV scale NP is appealing since they may be directly tested at colliders like the LHC and the other proposed ones through the search of the mode pp → + 1 + 2 + 4 jets [15]. In this latter case with |∆B| = 2 dinucleon decays, we noticed that there is a unique |∆B| = 2 single free or bound neutron decay mode n →pe + χ with χ being the standard model (SM) neutrino ν or anti-neutrinoν or any new light fermion beyond the minimal SM such as a light sterile neutrino, but such a mode has not yet been considered both theoretically and experimentally. In this letter we will investigate this neutron decay mode in the framework of effective field theory and point out its distinct experimental signature relative to other dinucleon decays for the guidance of future experimental searches. * hexg@phys.ntu.edu.tw † maxid@phys.ntu.edu.tw From the perspective of standard model effective field theory (SMEFT), the leading order interactions contributing to n →pe + χ depend on the types of χ. For the χ to be the electron neutrino ν e , n →pe + ν e can be generated, at leading order, by the same dimension-9 (dim-9) operators mediating the n−n oscillation through an insertion of a SM charged weak current, which is similar to the nuetron-antineutron conversion through the same n−n oscillation operators did by Gardner and Yan [16][17][18]. For the muon and tau neutrinos ν µ,τ , the leading order interactions for n →pe + ν µ,τ appear at dim 13, 1 this is because the SM has a good lepton flavor symmetry and therefore n →pe + ν µ,τ cannot be realized like the decay n →pe + ν e by dim-9 interactions.
On the other hand, for the antineutrino case, the decay mode n →pe +ν e,µ,τ also violates lepton number by two units (|∆L| = 2) and their leading order interactions appear at dim 12. After sending the Higgs field to its vacuum expectation value and expanding the quark and lepton doublet, we obtain the relevant effective interactions mediating n →pe + χ with χ = ν e in the so-called low energy effective field theory (LEFT) below the electroweak scale [20,21], which consisting of two up-type and four down-type quarks and a charged lepton current, i.e., having the configuration (uuddddeχ). For the decay n →pe + ν e , it is a little bit complicated since it can be generated at leading order from the same dim-9 n−n oscillation operators with the structure (uudddd) by inserting a SM four-fermion vertex in one of the four down-type quark legs.
Once the relevant LEFT operators are obtained, one can perform a non-perturbative QCD matching for the six-quark sectors using the baryon chiral perturbation theory (BχPT) formalism [22][23][24]. In this way, one ends up with operators consisting of nucleons, mesons and leptons and the decay rate can be readily calculated.
Here we note that the relevant interactions for pn → e + χ transition is the same as that of the decay n →pe + χ because these two processes are related to each other by crossing symmetry. Therefore constraints on the branching ratio of n →pe + χ can be obtained by using known bound from pn → e + χ transition. If realizing the interactions via SMEFT operators, these interactions may be related to other dinucleon to dilepton transitions such as pp → e + e + and nn →νν. An example of such a case will be given below. We will present the details elsewhere for the above procedures to our accompanying long paper concerning the |∆B = ∆L| = 2 dinucleon to dilepton decays (pp → + + , pn → +ν , nn →νν ) in a full EFT analysis [25].
EFT analysis for n →pe + χ. In the following, we will take the EFT procedures outlined in the above and start directly from the hadronic level interactions for a modelindependent analysis. We take the minimal assumption that all the dominant interactions contributing to n → pe + χ are encoded in the leading order dim-6 operators with the field configuration (npeχ).The relevant dim-6 effective |∆B| = 2 interactions are classified into two sectors in terms of the net lepton number |∆L| = 0 (for χ = ν) or |∆L| = 2 (for χ =ν) as follows where all the independent dim-6 operators in each sector are parametrized as follows where C is the charge conjugation matrix satisfying C T = C † = −C and C 2 = −1, and the flavor of SM lefthanded neutrino is suppressed. Except the neutron decay n →pe + χ, the above interactions can also contribute to the dinucleon decay pn → e + χ in nuclei and the conversion e − p → nχ and e − n →pχ in the electron-deuteron (e-d) scattering. This correlation of different processes due to the crossing symmetry can help us to estimate the partial lifetime of n →pe + χ from the experimental lower bound on that of pn → e + χ per oxygen nucleus 16 O set by the water Cherenkov Super-Kamiokande (SK) experiment [12]. In the following, we assume each time there is only one operator dominant for the decays n →pe + χ and np → e + χ, then the bound on the latter mode can be used to extract a bound on the former. As mentioned earlier that the low energy effective interactions may be related to NP at high energies, which can be parametrized through the SMEFT interactions. Here we give an example how a SMEFT operator can generate some of the low energy operators. We find the dim-12 operator O S,(S) is of a very interesting case. Where C is again the charge conjugation matrix, Q and L are the SM quark and lepton doublets, respectively. T SAA {mn}[ij][kl] = ijm kln + ijn klm is a color tensor responsible for the operator to be color invariant, ab ( ijk ) is the second (third) rank totally antisymmetric Levi-Civita symbol.
The hadronic counterpart of the six-quark part of O S,(S) Q 6 L 2 can be done through the BχPT matching, which has been used before for the dim-9 six-quark n−n oscillation operators in [24]. We first identify that the sixquark part of O S,(S) Q 6 L 2 belongs to the irreducible representation (3 L , 1 R ) of the two-flavor QCD chiral group SU (2) L × SU (2) R , then its leading oder hadronic counterpart is realized by appealing to the nucleon field Ψ = (p, n) T and pion matrix u in such a way that the resultant operator has the same symmetry property as O S,(S) Q 6 L 2 including the Lorentz covariance, baryon number and chiral property. After carrying out the procedures described in the above, we obtain the leading order hadronic operator taking the form ( where C S,(S) Here the hadronic low energy constants g 3×1 andĝ 3×1 are pertinent to the non-perturbative QCD matching of quark level operators in the BχPT. g 3×1 can be related to the n−n oscillation matrix element via chiral symmetry, the latter has been determined by using lattice QCD method [26], that implies g 3×1 ∼ 6 × 10 −6 GeV 6 , while the value forĝ 3×1 has not been calculated. A naive dimensional analysis would giveĝ 3×1 to be of order ∼ Λ 6 χ /(4π) 4 ∼ 1.2×10 −4 GeV 6 [27], where Λ χ ∼ 1.2 GeV is the chiral symmetry breaking scale. This result can also be estimated from the fact that the dim-9 six-quark operator condensates into a dim-3 nucleon current together with the couplingĝ 3×1 , the mass dimension of g 3×1 should be compensated by Λ QCD ∼ 200 MeV since it is the scale for the non-perturbative QCD effect, then g 3×1 ∼ Λ 6 QCD ∼ 6.4 × 10 −5 GeV 6 . We see both estimations agree within an O(10) uncertainty. For numerical estimations, we will use this later lower value for illustrations. The complete details for the above analysis will be presented in our accompanying long paper on LEFT and SMEFT operators [25].
It is also interesting that the above operator O S,(S) Q 6 L 2 not only generate n →pe +ν and np → +ν , but also pp → + + and nn →νν . If the future SK experiments could find the |∆B| = 2 signals from all those three different channels, it probably points to the NP incorporated in this operator.
Decay rate calculations. From the above interactions in Eqs. (1,2,3), it is straightforward to calculate the transition amplitudes for n(k) →p(p)e + (p 1 )χ(p 2 ) and henceforth the decay rates. The decay rate can be written as where m n is the neutron mass, the Mandelstam variables are defined as s = (p 1 + p 2 ) 2 , t = (k − p 1 ) 2 and u = (k − p 2 ) 2 .
Taking into the interactions in Eqs. (1,2,3) into consideration, after calculating the amplitudes and finishing the phase space integration, and assuming one term dominates at a time, we obtain the following decay rate for each type of the couplings, If several terms exist simultaneously, there are in general interference terms. In our numerical estimate later, we assume one term dominates at a time and therefore the interference terms are neglected. The dimensionless parameters i stem from the phase space integration over the kinematic variables and take the following numerical values where the suppression of i is because of the small phase space for the process. We now analyse the dinucleon to dilepton decay pn → e + χ in nucleus to obtain a correlation to the processes discussed above. From the same interactions in Eqs. (1,2,3), we can calculate the transition rate for pn → e + χ in nucleus in the following manner [28] where ρ p (k)(ρ n (k)) is the proton (neutron) density distribution in momentum space and ρ p (ρ n ) is the average proton (neutron) density. v 1 (v 2 ) is the velocity of the nucleon p(n) and v rel. is their relative velocity v rel. = |v 1 − v 2 |. The total cross-section for the free nucleon scattering process p(k 1 )n(k 2 ) → e + (p 1 )χ(p 2 ) takes the form where E 1 (E 2 ) is the energy of the initial state nucleon p(n). dΠ 2 is the relativistically invariant two-body phase space. For the oxygen nucleus 16 O, we neglect the small effect due to the effects of nucleon Fermi motion and nuclear binding energy [29]. Under the quasi-static approximation of nucleons, then the transition rate reduces into where the average nuclear matter density ρ N approximately equals 0.25 fm −3 for either proton or neutron, and m N = (m n + m p )/2. The two-body final state phase factor Π 2 = (1/8π) 1 − m 2 e /(m n + m p ) 2 . The relevant amplitudes are calculated from the interactions in Eqs. (1,2,3). After finishing the phase space integration and combining all pieces together, we obtain where again we neglect the interference terms between any pair of Wilson coefficients under the assumption of one operator dominant each time. η i are dimensionless parameters and defined as where δ = m 2 e /m 2 N and the nucleon velocity effect is neglected. If we keep the small nucleon velocity effect, η 1,3 will be modified to be proportional to the squared velocity, i.e., η 1,3 ∼ v 2 N ∼ 2∆E/m N ∼ 10 −2 . The nucleon velocity v N is estimated to be around 0.1 c by taking the average nucleon binding energy ∆E ∼ 8 MeV in oxygen nucleus, where c is the speed of light. For η 2,4,5 , the nucleon velocity effect is small relative to their leading contribution and can be safely neglected. In the following, we use the modified value η 1 ∼ v 2 N ∼ 10 −2 for η 1 but still keep η 3 's value as a conservative estimation for the bound on partial lifetime of n →pe + χ.
In order to employ the experimental result for np → e + χ, we assume each time there is only one operator dominant. According to the decay rate in Eq. (7) for n →pe + χ and Eq. (12) for np → e + χ, after keeping one term each time and eliminating the common Wilson coefficient, we obtain the following relations where in the last step we take the SK lower bound on the partial lifetime of pn → e + χ as our input, in which Γ −1,SK pn→e + χ 2.6 × 10 32 yrs [12]. Then our main result for the prediction of the |∆B| = 2 neutron decay mode n →pe + χ is as follows we see that the extrapolated lower bound on the partial lifetime varies from O(10 39 ) to O(10 53 ) years depending on which operator is dominant. Furthermore, if we take η 3 ∼ 10 −2 as did for η 1 , the lower bound is improved to be O(10 45 ) years. As a comparison with the experimental bound on the inclusive mode Γ −1,PDG n→e + anything 0.6 × 10 30 yrs quoted by the Particle Data Group [30], we find the indirect bound obtained here are improved by at least ten orders of magnitude.
Even though we obtained the above bound on the SM neutrino case, the bound is also valid for beyond the SM light fermion with a mass m χ < m n − m p − m e , the analysis is similar to the above SM neutrino case. If the mass m χ = 0, we would expect a stronger bound on the partial lifetime due to an even smaller phase space.
We can also use the known constraints on dinucleon to dilepton transition rate to set bound on NP scale. Applying the SK experimental limits on the transitions (pp → + + , pn → +ν , nn →νν ) from our example operator O S,(S) Q 6 L 2 [12,14], we find the scale of NP Λ NP 2 − 3 TeV. This may be probed at the LHC by looking at the process pp → + + + 4 jets or the future LHeC via e − p → + + 5 jets.
Discussions. From our analysis in this paper, we expect the partial lifetime for the unique |∆B| = 2 neutron decay mode n →pe + χ is far longer than the current experimental sensitivity. Nevertheless, this decay mode has several distinctive features in experimental searches and also strong astrophysical implications. First, in the bound nuclei, if a neutron decays into an antiproton, then the antiproton could annihilate with a neighbour proton to release two bunch of energetic mesons in the opposite direction through the QCD interactions, or two O(1 GeV) gamma photons via the QED interaction, such signals are clear evidence of this decay mode if they could happen. They may be searched for in the future neutrino experiments like the Super-K, DUNE [31], JUNO, etc. Second, unlike the |∆B| = 2 dinucleon to dimeson/dilepton decays [12][13][14], this decay mode can take place for the free neutron and it opens up the possibility to search this |∆B| = 2 signal from free neutron decay and oscillation experiments like the European Spallation Source [32]. This neutron decay mode could also have impact on the astrophysical processes like the cooling of neutron star and the evolution of the universe, which can be used to constrain the relevant NP complementarily.