$n$-$\bar n$ oscillations and the neutron lifetime

Neutron-antineutron oscillations are considered in the light of recently proposed particle models, which claim to resolve the neutron lifetime anomaly, indicating the existence of baryon violating $\Delta B=1$ interactions. Possible constraints are derived coming from the non-observation of neutron-antineutron oscillations, which can take place if the dark matter particle produced in neutron decay happens to be a Majorana fermion. It is shown that this can be realised in a simple MSSM extention where only the baryon number violating term $u^cd^cd^c$ is included whilst all other R-parity violating terms are prevented to avoid rapid proton decay. It is demostrated how this scenario can be implemented in a string motivated GUT broken to MSSM by fluxes.


Motivation and facts
Neutrons, together with protons and electrons are the fundamental constituents of atomic matter and their properties have been studied for almost a century. A free neutron, in particular, disintegrates to a proton, an electron and its corresponding antineutrino, according to the well known β-decay process n → p + e − +ν e . Notwithstanding those well known facts, the precise lifetime of the neutron remains a riddle wrapped up in an enigma. The problem lies in the fact that the two distinct techniques employed to measure the lifetime end up to a glaring discrepancy [1]. More specifically, in one method a certain number of neutrons are collected in a container [2] (known as "bottle"), where, after a certain time duration (comparable to the neutron lifetime), several of them decay. The remaining fraction of them can be used to determine the lifetime which is found to be τ n ≈ 879.6 ± 0.6 sec. In an alternative way of measuring the lifetime named [3], [4] "beam", a neutron beam with known intensity is directed to an electromagnetic trap. Counting the emerging protons within a certain time interval, it is found that their numbers are consistent with a neutron lifetime of τ n = 888.0±2.0 sec. These two measurements display a 4.0 σ discrepancy which cannot be attributed to statistical uncertainties. An explanation of this difference of the two measurements could be that other decay channels contribute to the total lifetime in the "beam" case. In the context of the minimal Standard Model, however, there are no available couplings and particles that lead to such a channel and could thus account for this difference. According to a recent proposal [5] the discrepancy could be interpreted if neutrons have a decay channel to a dark matter (DM) candidate particle χ with a branching ratio ∼ 1% and a mass comparable to the neutron's mass. The simplest possibility is realised with the neutron decay to a two particle final state consisting of a DM fermion χ and a monochromatic photon, n → χ + γ. Operators describing this type of decays, however, violate baryon number. In the microscopic level, the description of the above decay requires the existence of a colour scalar field with the quantum numbers of a Standard Model (SM) colour triplet, D = (3, 1) −1/3 , with mass M D ≥ 1 TeV and the couplings Two basic assumptions have been made for this scenario to work. Firstly, it is assumed that other baryon violating couplings of the new colour triplet, D = (3, 1) −1/3 , are substantially suppressed. Indeed, a colour triplet, introduces other baryon non-conserving couplings similar to the R-parity violating (/ R) ones of the supersymmetric theories. Unless their couplings are unnaturally tiny, they lead to fast proton decay at unacceptable rates. In the context of SM, there are no obvious symmetries which prevent their appearance while leaving the terms (1) intact. Secondly, it is assumed that the DM fermion χ is a Dirac particle. Since χ is a neutral field, however, it could be likewise a Majorana particle and, in such a case, might contribute to n-n oscillations.
In this letter, we reconsider the interpretation of the neutron lifetime discrepancy described above, in the context of Minimal Supersymmetric Standard Model (MSSM) extensions and in particular SUSY and String motivated GUTs. There are many good reasons to implement the above scenario in this context. We firstly remark that the kind of the scalar particle introduced to realise the processes has the quantum numbers of a down quark colour triplet. Thus, in the context of MSSM, this could be the scalar componentd c j of a down quark supermultiplet. There are good chances that the supersymmetry breaking scale is around the TeV scale and the sparticle spectrum may be accessible either at the LHC or its upgrades. Thus, taking into account the recent bounds of LHC experiments, its mass mdc j could be around the TeV scale which is adequate to interpret the neutron lifetime discrepancy. Notice however, that in the MSSM context, terms such as (1) appear together with other baryon and lepton number violating interactions giving rise to fast proton decay, and therefore, they are forbidden by Rsymmetry. There are examples of Grand Unified Theories with string origin, however, equipped with symmetries and novel symmetry breaking mechanisms where it is possible to end up with a lagrangian only with the desired / R-coupling and all the others forbidden. Thus, in the presence only of the trilinear coupling shown in (1) which can account for the discrepancy, the only possible baryon violating processes is neutron-antineutron oscillations. Our aim in the present work, is to investigate under what conditions the issue of neutron lifetime is solved. In particular, we will examine whether the strength of the couplings and the mass scale required to interpret the discrepancy, are consistent with the bounds on n-n oscillations.
The layout of the paper is as follows: In section 2 we present a short overview of gauge invariant baryon and lepton number violating symmetries in the context of GUTs with emphasis on R-parity violating supersymmetry. In section 3 we summarise the essential formalism related to neutron-antineutron oscillations and in section 4 we present the main results, including bounds of the relevant baryon violating couplings in the TeV scale extracted from the current limits of n-n oscillations. Some concluding remarks and a short discussion are presented in section 5. Finally, for the readers convenience, some detailed formulas entering our calculations are given in the appendix.

A brief overview of R-parity in fluxed GUTs
In the non-supersymmetric standard model, at the renormalisable level, Baryon (B) and Lepton (L) numbers are conserved quantum numbers, due to accidental global symmetries. This fact is consistent with the observed stability of the proton and the absence of lepton decays (such as ββ-decay) which violate B and L. Introducing new coloured particles which imply additional interactions, however, this is no longer true.
In the supersymmetric lagrangian of the Standard Model symmetry, in principle, one could write down gauge invariant terms which violate B and L mumbers. In superfield notation these are: If all these couplings were present, for natural values of Yukawas λ ijk ∼ O(10 −1 ), violation of B and L would occur at unacceptable rates. As is well known, in the minimum supersymmetric standard model the adoption of R-symmetry prevents all these terms.
Without the existence of R-symmetry or other possible discrete and U (1) factors, these terms are also present in GUTs. In the minimal SU (5) for example, the most common B and L violating terms arise from the coupling In a wide class of string motivated GUTs there are cases where some of the terms in (3) are absent in a natural way. In a particular class of such models, where the breaking of the gauge symmetry occurs due to fluxes which are switched on along the dimensions of the compact manifold, we may have for example the following SM decomposition: which is just the operator required to mediate n-n oscillations. The absence of the remaining couplings in (2) ensures that the proton remains stable, or its decay occurs at higher orders in perturbation theory and therefore its decay rate is highly suppressed and undetectable from present day experiments.
To be more precise, focusing in SU (5) as a prototype unified theory, the flux mechanism works as follows [6]: Assuming that SU (5) chirality has been obtained by fluxes associated with abelian factors embedded together with SU (5) into a higher symmetry, another flux is introduced along the hypercharge generator U (1) Y to break SU (5) GU T [6]. It turns out that this is also responsible for the splitting of the SU (5) representations. If some integers M, N represent these two kinds of fluxes piercing certain "matter curves" of the compact manifold hosting the 10-plets and 5-plets, the following splittings of the corresponding representations occur: The integers M 10 , M 5 , N are related to specific choices of the fluxes, and may take any positive or negative value, leading to different number of SM representations. Hence, there is a variety of possibilities which can be fixed only if certain string boundary conditions have been chosen. In order to exemplify the effect of these choices, here we assume only a few arbitrary cases where the integers M, N take the lower possible values ±1, 0. 3 for the SU (5) representations. Substituting these numbers in (5,6) we obtain a variety of possibilities, and some of them are shown in Table 1.

10-plets Flux Units
10 Content 5-plets Flux Units5 Content Hence, we end up with incomplete SU (5) representations. Some examples of R-parity violating operators formed by trilinear terms involving the above incomplete representations are shown in Table 2. (for a comprehensive analysis and a complete list of possibilities see [7]). We Table 2: SU (5)-fluxed representations with incompleted MSSM content, / R-processes emerging from the trilinear coupling 10 i5j5j for selected combinations of the multiplets given in Table 1. observe that the couplings 10 1 ·5 5 ·5 5 and 10 3 ·5 5 ·5 5 in the last two lines of this table give exactly the required R-violating trilinear coupling, while all the other couplings are absent. This is just the case that will be considered in the subsequent analysis.

neutron-antineutron oscillation formalism
In this section we will briefly present the main features of the n-n oscillations mainly to establish notation and put the recently baryon violating scenario, proposed for the extra exotic channel of neutron decay to a light dark matter, in a broader perspective. In this context additional processes entering n-n oscillations at tree level or at the one loop level are presented.

neutron and antineutron bound wavefunctions
We will consider the neutron as a bound state of three quarks (antiquarks) for neutron (antineutron), in a colour singlet s-state in momentum space. The orbital part is of the form: where P is the hadron momentum and: with p i , i = 1, 2, 3 the quark momenta. The functions φ(ξ), φ(η) are assumed to be 0s harmonic oscillator wavefunctions. These functions are assumed to be normalised the usual way:

neutron-antineutron transition mediated by dark matter Majorana fermion
A dark matter colourless Majorana fermion of mass m χ emitted from a neutron of momentum P and absorbed by an antineutron of momentum P ′ can lead to n-n oscillations. This process is exhibited in Fig. 1(a). In a previous study [5] this did not happen, since the mediating fermion was assumed to be a Dirac like particle, but there is no reason to restrict in this choice. In fact there exists the possibility of this particle being a Majorana like fermion in which case neutron-antineutron oscillations become possible.
The orbital matrix element takes the form: Using (9) the matrix element can be written as follows: Now the 0s wavefunction is Thus, performing the Gaussian integral, we get , and the orbital part becomes It is instructive to compare this with the probability for finding the quark at the origin inside the nucleon: The colour factor is quite simple since it involves the same hadron. It takes the form: where (ud) S (0, 1) −α is the flavor symmetric colour antisymmetric two quark state, φ α the conjugation phase, and (0,0) is the colour singlet hadronic state. Thus The factor of 1 2 came from chirality since the propagating fermion is only left-handed. In the case of supersymmetry induced oscillation, Fig. 1(b), we find an analogous expression. The orbital part is similar to the previous one with the obvious modifications m D → md, λ q λ χ → c bd λ ′′ ubd g/2. Here, c bd is the flavor violating mixing between the scalarsd andb c , which induces baryon violation. Thus: Including the colour helicity factors we set

Additional neutron-antineutron mechanisms at tree level
n-n oscillations with gluino exchange take place at tree level, see fig 2. This is directly comparable with n-decay process through DM particle χ. However, because of the colour antisymmetry, the coupling u c d cdc cannot be realised directly and it requires mass insertion, thus a suppression factor emerges due to assumed mixing betweenb c ,s c and their left componentsb,s. (This requirement is beyond the minimal flavour violation scenario which assumes diagonal mass matrix.) It is known that di-nucleon decays to two Kaons, N N → K K, impose stringent constraints on the coupling λ ′′  The gluino exchange diagram of Fig. 1(c), (see also Fig. 2), differs from that of Fig. 1(b) in the 4 See for example [8,9] and references therein.
The colour factor is a bit more complicated. We encounter the combination: with a similar combination on the other hadron. The states are specified as follows: in the standard SU(3) labeling of the states [10] |(λ, µ)ǫ, Λ, Λ 0 .
The allowed by SU(3) symmetry coefficients can be easily calculated from the tables involving the reduction (01)⊗(10) → (11), table 1 of ref. [11]. The obtained results are presented in table 3. Expanding the hadronic states in terms of an antisymmetric pair of quarks and a single quark we find and thus, we get: In this case there is no need to have flavour off diagonal baryon violating interactions, a mixing between the scalarsb andb c is adequate. The generation mixing can be induced as in the standard model via the Wino and the W-boson in a box diagram. In this case the interaction between the neutron and antineutron does not take the simple form found above at tree level. Since, however, theb-scalars are quite heavy, it leads to a contact interaction, see Fig. 1(d). Since no colour particle propagates between the two hadrons the colour factor is 3 and the orbital part can be written in the form:

neutron-antineutron transition mediated by box diagrams
where s box is dependent on the masses of the particles circulating in the loop, namely the Wboson, the top quark, the Wino and theũ scalars. Thus where g 2 = 4 √ 2G F m 2 W ≈ 0.4 and s box will be evaluated in Appendix A.

neutron-antineutron oscillation results
Combining the two cases, namely the non-supersymmetric dark matter and the corresponding supersymmetric processes, we find that the transition amplitude takes the form: where s box ≈ 3.0 × 10 −6 , see the Appendix A. Due to this factor as well as the small mixing c bb , the parameter λ ′′ udb need not be extremely small. Notice also that a graph involving the bino will give a contribution similar to the second term. Analogous graphs involving Higgsinos are also possible, but they provide no new insights and will not be elaborated.
It is now natural to assume that the mass of the propagating scalar is the same in all models. If constraints come from other experiments we will compensate by adjusting the relevant couplings. Then we can take as scale of the masses to be of the order 1 TeV. Another parameter to be determined is the nucleon size parameter which is usually taken to be 0.8 fm. This is related to the nucleon wavefunction at the origin In reference [5] the value of ψ(0) 2 = 0.014 GeV 3 was adopted taking into account effects arising from lattice gauge calculations [12]. This leads to a value of about 0.5 fm. We will adopt this value in the present calculation. Thus we can write κ in the form: with The n-n mixing matrix becomes m n 1 κ κ 1 , which leads to complete mixing with energies E 1 = m n (1 + κ), E 2 = m n (1 − κ). Thus the neutron-antineutron oscillation probability in vacuum becomes In other words the oscillation time is: In the presence of matter the diagonal elements of the matrix are not the same, since the neutron and the antineutron interact differently with any surrounding magnetic field or matter. A tiny magnetic field of the order of 10 −10 T can lead to an energy difference of ∼ 10 −26 m n . In current experiments the magnetic fields are limited [13] to 10 −8 T, which leads δ ≈ 10 −24 . Thus the oscillation probability becomes [13] P (n ↔n) = (κ/δ) 2 1 + (κ/δ) 2 sin 2 (m n δ 2 + κ 2 t)e −λt ≈ κ 2 δ 2 sin 2 (m n δ 2 + κ 2 t) e −λt · where λ = 1/τ n with τ n the neutron life time, as e.g., measured in the "bottle" experiment mentioned in the introduction. The non-observation of neutrino oscillations implies κ ≪ δ, κ < 10 −9 δ = 10 −33 , τ > 10 8 s.
In the context of the R-parity violating supersymmetry model [8] we will try to extract a limit on the value of c bd λ udb in the case of the tree diagrams and the c bb λ udb in the case of the box diagram from the non observation of n-n oscillation, i.e., solve the relations: We will consider each case separately: • Gluino exchange. Taking α s = 1 and mg = 500 GeV we obtain: • A SUSY dark matter particle (W 3 orB), exchange (see fig. 1(d)(b)). Taking mW 3 = 500 GeV we obtain: |c bd λ udb | 2.0 × 10 −7 · (33) • Finally in the case of the box diagram taking s box = 3.0 × 10 −6 and mW = 500 GeV we obtain a weaker upper bound of the order

Discussion
In a recent paper [5] a very interesting proposal was made to resolve the long standing discrepancy on the determination of neutron lifetime measured in experiments involving trapped neutrons in a "bottle" and neutrons decaying in flight ("beam" experiments). This model considers novel mechanisms for neutron decays involving new dark decay channels in the "bottle" case, where the decay products contain light dark matter particles, with mass in a slim range between the neutron and proton mass. The final state of this reaction might also involve visible particles such as photons. These scenarios sparked off a renewed activity on this issue and astrophysical as well as experimental constraints on the various decay modes have been discussed. Hence, in a recent analysis [14] decay channels involving a light dark matter particle and a visible photon were ruled out, while decays involving dark photons are subject to stringent constraints from astrophysical observations [15]. Furthermore, it has been suggested [16] (see also [17]) that neutron decay to dark matter is in conflict with neutron stars, but the argument does not involve free neutrons.
In the present work, we have explored two different aspects of this proposal, namely the implications on baryon number violation and the possible Majorana nature of the emitted dark matter light particle.
We firstly focused on the fact that this decay process is realised with the mediation of colour triplets. In the context of the Standard Model and its obvious supersymmetric extensions, such particles generate other dangerous baryon and lepton number violating interactions, unless their coupling strengths to ordinary matter are unnaturally small. We have suggested that this problem can be remedied in the case of a class of SUSY GUTs derived in the framework of string theories where "fluxes" developed along the compact dimensions are capable of eliminating the superpotential terms associated with the undesired interactions.
Furthermore, we have considered the possibility that the neutral dark matter particle in the putative exotic neutron decay channel is of Majorana type. In this case we find, however, that the parameters employed in this model are in conflict with the neutron-antineutron oscillation limits. We have considered limits from such baryon number violating processes in the context of R-parity violating supersymmetry, both at tree as well as at the one-loop order. We find the most stringent limit on the parameter |c bd λ udb | ≤ 1.2 × 10 −8 comes from gluino exchange. The weaker limit of |c bb λ udb | 10 −4 comes from the box diagram. The difference can be attributed to the fact that the tree diagrams involve both baryon and family flavor change of the participating s-quarks, while the loop diagram is diagonal in flavor.
GKL would like to thank Maria Vittoria Garzelli at UDEL for useful suggestions.

Appendix A: The box contribution
For the non expert reader we provide some details regarding the evaluation of the box diagram contribution.
The gluino exchange diagram requires non-minimal mixing which might not be present in simple supersymmetric models. Hence, we assume the case whereb L ,b R have a non-trivial mixing term where A ef f = A−µ tan β, A being a soft SUSY breaking parameter, µ the Higgs mixing (µ-term) and tan β the Higgs vev ratio. Similar terms can exist for the other two families. As a result, the process receives contributions from one-loop box graphs involving Winos. This is depicted in fig 3. The possible R-parity violating terms contributing to the process are λ ′′ udb u c d c b c and λ ′′ uds u c d c s c . Only λ ′′ udb is shown in the figure since, as explained above, λ ′′ uds is suppressed. Moreover, due to the larger b-quark mass m b compared to m s , factors such as m 2 b /m 2 W , enhance the effect.
The processes requires the sequence of reactions: Initially d R u R + d L → d L +b * R followed by d L +b * L →b L +d L , from the W -boson and Wino exchange box diagram. At the final stage we getd L +d RūR .
Calculation of the diagram gives the following relation for the decay rate [8] with m 2 b LR given by (35) and ξ jk being the following combination of CKM matrix parameters: The computation of the loop integral in (36) is parametrised by the function Ω which depends on the four masses circulating in the box and is given by: The current experimental lower bound on n-n oscillation period τ = 1 Γ is τ 10 8 s [13], (see section 4).
In our notation |ψ(0)| 2 is the baryonic wavefunction matrix element for three quarks inside a nucleon estimated [5,12] to be |ψ(0)| 2 = 0.014 GeV 3 . From eq.( 36) we can recalculate the bounds on λ ′′ udbb coupling using the latest LHC bounds on scalar masses involved in the box graph. However, knowing only the lower bounds on this large number of arbitrary mass parameters through this complicated formula, is not very illuminating. Thus, before going to the most general case, in order to reduce the number of arbitrary mass parameters, and have a feeling of the contributions of the various components, we first examine the limit mũ → mW The remaining contributions (40) display a complicated dependence on the SUSY scalar masses, but for masses close to the experimental lower bounds, they can be of the same order. In such cases, depending on the signs of the CKM mixing parameters ξ jk there might be cancellations which result to weaker bounds on the λ ′′ udb couplings. To examine this general case, we use the Equation (36) to recalculate the bounds on λ ′′ udb taken into account the latest experimental results for the SUSY mass parameters. We take A ef f = 400 GeV which fixes the ratio (42) to be c bb ∼ 10 −2 .
In Figure 4 we fix mb L = mb R = 450 GeV. The three curves correspond to stop masses of 450, 625 and 750 GeV. As we can see, leaving aside accidental cancellations, the value of λ ′′ udb is constrained to be less that ∼ 0.15 − 0.3.
We will now estimate s box making use of Eq. (12) by writing: assuming the range mw ≈ 400-500 GeV. We have, of course, removed the factors 3, (1/2), g 4 and the scalar masses from the expression of Eq. (41) since they appear explicitly in Eq (23).