Antinucleon-nucleon interaction from chiral effective ﬁeld theory and its application to neutron-antineutron oscillations

. An ¯ NN potential is reviewed which has been derived within chiral e ﬀ ective ﬁeld theory by the Jülich-Bonn group. By construction it reproduces ¯ NN phase shifts and inelasticities, provided by an elaborate phase-shift analysis of available ¯ pp scattering data. This interaction is then employed in a study of neutron-antineutron oscillations in the deuteron. In particular, the deuteron lifetime is considered, and results for it are presented in terms of the free-space n − ¯ n oscillation time, based on that ¯ NN potential in combination with an NN interaction likewise derived within chiral e ﬀ ective ﬁeld theory.


Introduction
In Ref. [1] a new generation of nucleon-nucleon (NN) potentials was derived in the framework of chiral effective field theory (EFT).In that publication a novel local regularization scheme was introduced and applied to the pion-exchange contributions to the NN force.Furthermore, an alternative scheme for estimating the theoretical uncertainty was proposed that no longer depends on a variation of the cutoff [2,3].In a recent study the Jülich-Bonn group took advantage of this development and adapted the formalism and the concepts proposed by Epelbaum et al. in order to establish a high-precision antinucleon-nucleon ( NN) potential within chiral EFT [4].Specifically, in that work an NN potential up to next-to-next-to-nextto-leading order (N 3 LO) in the perturbative expansion was presented, which extends a previous work by the Jülich-Bonn group that had considered the NN force only up to N 2 LO [5].In both cases the strength parameters of the contact terms that arise in the EFT framework [3,6] (i.e. the low-energy constants or simply LECs) have been fixed by a fit to the phase shifts and inelasticities provided by a proper phase-shift analysis of pp scattering data [7].
The NN potentials from chiral EFT have been already successfully used in studies of the electromagnetic form factors of the proton in the time-like region [8] and of the unusually large near-threshold enhancement in the pp spectrum detected in the reaction J/ψ → γ pp [9].A further and certainly more exotic application of this NN interaction has been presented by us in Ref. [10], where neutron-antineutron oscillations in the deuteron have been studied.
Neutron-antineutron (n − n) oscillations involve a change of the baryon number (B) by two units (|∆B| = 2).An experimental observation would allow a glimpse on physics beyond the standard model, see e.g.[11][12][13].Since in such oscillations B is violated the process satisfies one of the Sakharov conditions [14] that have been formulated in order to explain the observation that there is more matter than anti-matter in the universe [15].
The key quanitity in this subject is the free n − n oscillation time, τ n−n .The presently best experimental limit on it is τ n−n > 0.86 × 10 8 s ≈ 2.7 yr (with 90 % C.L.) [16].Additional information can be deduced from studies of n − n oscillations in a nuclear environment.Corresponding experiments have been performed, e.g., for 56 Fe [17], 16 O [18], and for the deuteron ( 2 H) [19], while others are planned [20,21] In such a case the oscillation process is suppressed as compared to the free situation.The pertinent lifetime τ nuc is commonly expressed in terms of the one in free space as [12] where R is an intranuclear suppression factor, also called reduced lifetime, that depends on the specific nucleus.It can be calculated from nuclear theory and then can be used to relate the measured lifetimes of those nuclei with the free n − n oscillation time [12], see, e.g., Refs.[22][23][24][25][26].
For a long time the suppression factors published in 1983 [23] have been used as standard by experimentalists in the interpretation of their measurements [17,19].For example, in case of the deuteron the corresponding value is R ∼ (2.40 − 2.56) × 10 22 s −1 , a prediction based on the phenomenological antinucleon-nucleon ( NN) potentials by Dover and Richard [27,28].Recently, however, those values have been challenged in a work by Oosterhof et al. [29].In that study an effective field theory for the |∆B| = 2 interaction is constructed and the quantity R is evaluated within the power counting scheme proposed by Kaplan, Savage, and Wise (KSW) [30,31] for the nucleon-nucleon (NN) and NN interactions.The value of R for the deuteron obtained in that approach is (1.1 ± 0.3) × 10 22 s −1 , about a factor 2 smaller than the one by Dover et al. [23].
In the light of this controversal situation a new calculation of the suppression factor for the deuteron has been performed by us [10].It was prompted by the aim to utilize the discussed modern chiral interactions for the involved NN [1] and NN [4] systems.Specifically, in case of the NN interaction most of the available precise pp scattering data (mostly from the LEAR facility at CERN [32]) have appeared only after the publication of the potentials used in Ref. [23].Therefore, an update is long overdue.Of course, the main motivation was the aforementiond discrepancy reported in Ref. [29] and the prospect to find a plausible explanation for that difference.
The paper is structured in the following way: In Sect. 2 the employed NN potential is described and some results for np, relevant for the calculation of n − n oscillations in the deuteron, are presented.In Sect. 3 a basic description of the formalism for evaluating the n − n oscillations in the deuteron is provided.Results for the oscillations, specifically for the suppression factor R, are presented in Sect. 4. The paper closes with a brief summary.

The NN interaction in chiral EFT
The derivation of the chiral NN potential using the Weinberg power counting is described extensively in Ref. [4], and we refer the reader to this work for the details.As already indicated above, the chiral potential contains pion exchanges and a series of contact interactions with an increasing number of derivatives.Up to N 3 LO there are contributions from one-, twoand three-pion exchanges.Those are identical to the ones that appear in the NN potential [1].However, there is a sign change in case of an odd number of exchanged pions due to its negative G parity, i.e.V NN = −V π + V 2π − V 3π , with V π , etc., the corresponding contributions to the NN force.On the other hand, the contact interaction, V NN cont , cannot be taken over simply from the NN case.Those contact terms represent effectively the short-range part of the interaction and, therefore, the G parity of the indivudal contributions remains unresolved.Thus, the strength parameters associated with the arising contact terms, the LECs, need to be determined in a fit to NN data.We fix them by fitting to the phase shifts and inelasticity parameters of the PWA of Zhou and Timmermans [7].How this is done is described in detail in Ref. [4].Note that there are more independent LECs in the NN case than in NN because in the former system there is no restriction from the Pauli principle.In addition, in our approach NN annihilation (into multi-meson channels) is likewise parameterized by contact terms, see Refs.[4,5] for explicit expressions.
Once the potential is established, the reaction amplitude is obtained from the solution of a relativistic Lippmann-Schwinger (LS) equation.It reads in partial-wave projected form Here p and p are the (moduli of the) center-of-mass NN momenta in the final and initial states, respectively, and where k is the on-shell momentum.We adopt a relativistic scattering equation so that our amplitudes fulfill the relativistic unitarity condition at any order, as done also in the NN sector [2,6].On the other hand, relativistic corrections to the potential are calculated order by order.They appear first at N 3 LO in the Weinberg scheme, see Appendix A in Ref. [4].L, L , etc., specifies the orbital angular momentum, considering that the NN system can be in uncoupled (spin-singlet or triplet) states where L = L = L = J or in coupled partial waves with L , L , L = J − 1, J + 1.
Since the integral in the LS equation ( 2) is divergent for the chiral potentials [2,6] a regularization needs to be introduced.For that the regularization scheme of Ref. [1] is utilized, where a local regulator is used for the pion-exchange contributions and a nonlocal regulator for the contact terms: The explict form of the regulation functions is f R [1] where Λ = 2/R is used for relating the momentum-space cutoff parameter with the cutoff radius.
A complete overview of our results for NN scattering up to N 3 LO in chiral EFT can be found in Ref. [4] while possible NN bound states are discussed in Ref. [33].Here we focus on quantities that are relevant for the study of n − n oscillations in the deuteron.Specifically, we look at the 3 S 1 -3 D 1 phase shifts in the np (isospin I = 1) channel (Fig. 1) and the np cross section, see Fig. 2. Note that data for the latter reaction were not included in the PWA [7] because they are less precise than the ones for pp.The bands in Figs. 1 and 2 represent the estimated uncertainty of our results.Here we follow the suggestion of Ref. [1] and use as measure the expected size of the higher-order corrections together with the actual size of the higher-order corrections, see Ref. [4] for details.
In Fig. 3 we present the np annihilation cross section multiplied by the velocity β of the incoming n.This allows a more detailed view on the situation very close to the threshold.We consider only the N 3 LO interaction with cutoff radius R = 0.9 fm because the theoretical NN phase shifts 7(2017)078 nd imaginary parts of NN phase shifts for the potential with cutoff R = 0.9 fm.cribed in the text.The filled circles represent the solution of the pp PWA [32].
-18 - The dashed line is the contribution from the 3 S 1 -3 D 1 partial wave while the solid line is the full result.Data are from Refs.[34,36].
uncertainty at low momenta is very small.One can see that most of the contributions come indeed from the 3 S 1 -3 D 1 partial wave, that is relevant for the evaluation of n − n oscillations.
The figure reveals also that the data from the OBELIX Collaboration [36] do not follow the expected near-threshold behavior.The cross sections for exothermic reactions should behave like 1/β so that β σ ann is then approximately constant for low momenta.This expectation is well satisfied by the BNL data [34], but also by the theory predictions.A discussion of the anomalous behavior suggested by the OBELIX data can be found in Ref. [35].
In the study of n − n oscillations below we employ also an earlier (N 2 LO) NN potential published by our group.A description of that interaction can be found in Ref. [5].We consider the two NN interactions because they are based on rather different regularization schemes.In the earlier potential [5] a non-local exponential exponential regulator was employed for the whole potential while, as outlined above, in the N 3 LO interaction [4] a local regulator was adopted for the evaluation of the one-and two-pion contributions.Comparing the pertinent results allows us to shed light on the question in how far the choice of the regulator influences the predictions.
For the needed deuteron wave function we take those from the NN potentials derived in the same framework and at the same order.In particular, for the calculation with the new NN interaction [4] we take the wave functions from Ref. [1], whereas for that with the (N 2 LO) NN potential [5] the wave functions from Ref. [2] are utilized.For exploring the sensitivity of the results to the deuteron wave function we employ also those from two meson-exchange potentials [37,38].

Formalism for n − n oscillations
For evaluating the n − n oscillations in the deuteron we follow the formalism presented in Refs.[22,23].The starting point is the eigenvalue (Schrödinger) equation [22] EPJ Web of Conferences 262, 01017 (2022) https://doi.org/10.1051/epjconf/202226201017EXA2021 Here, V np and V np are the potentials in the np and np systems and |ψ np and |ψ np are the corresponding wave functions.The systems are coupled via V n−n which is given by the n − n transition matrix element δm n−n where the latter is proportional to the inverse of the n − n oscillation time, i.e.V n−n = δm n−n = /τ n−n [12].
To leading order the np component |ψ np obeys the equation where E d is the unperturbed energy of the deuteron and |ψ d is the corresponding deuteron wave function.The decay width of the deuteron, Γ d , is then [22] We solve Eq. ( 5) in momentum space.Performing a partial wave decomposition and taking into account the coupling of the 3 S 1 and 3 D 1 channels, the above integral equation reads with L, L = 0, 2. Note that E d is the total energy corresponding to the deuteron, i.e.
and the width is provided by The deuteron lifetime τ d is given by τ d = /Γ d .The interesting quantity is the so-called reduced lifetime R [22,23,26] which relates the free n − n oscillation lifetime with that of the deuteron,

Results and discussion
Our results for the reduced lifetime R for the deuteron are summarized in Table 1.They are based on our N 3 LO interaction with cutoff R 0 = 0.9 fm [4] and the N 2 LO interaction with cutoff {Λ, Λ} = {450, 500} MeV [5].For technical details see the corresponding publications.
Besides the predictions for R based on the chiral NN interactions we list also the values given in Ref. [23] where the NN potentials DR 1 and DR 2 by Dover-Richard [27,28] have been utilized.Furthermore we include results from the calculation of Oosterhof et al. performed directly within EFT on the basis of the KSW approach.In this case R can be represented in a compact analytical form which reads up to NLO [29] Obviously, the only parameter here is the np 3 S 1 scattering length.All other quantities that enter are well established NN observables, cf.Ref. [29] for details.Note that in that paper, the scattering length Re a np was taken from Ref. [4].
Table 1.Reduced lifetime R calculated for the χEFT NN potentials from Refs.[4,5], together with information on the pertinent np 3 S 1 scattering length.Results for the Dover-Richard potentials DR 1 and DR 2 are taken from Ref. [23].The corresponding scattering lengths are from Ref. [39].Predictions based on Eq. ( 11), i.e. on the KSW approach applied in Ref. [29], are indicated too.
χEFT N 2 LO [5] χEFT  1 reveals that the values for R predicted by the chiral NN interactions are fairly similar to those obtained for the DR potentials in the past.By contrast, the results based on the framework employed by Oosterhof et al. [29] are rather different.Since the scattering length from the N 3 LO chiral NN interaction [4] is utilized in that work, the large discrepancy observed in Ref. [29] is certainly not due to differences in Im a np but must be primarily a consequence of the different approaches.
For investigating the sensitivity of our results to the used ingredients we performed various exploratory calculations.Specifically, we employed the NLO and N 2 LO variants of the considered NN (and NN) interactions.The corresponding predictions for R were found to lie within a range of (2.48 − 2.65) × 10 22 s −1 .If one takes this variation as measure for the uncertainty due to the nuclear structure, i.e. the NN and NN interactions (wave functions), a value of roughly R = (2.6 ± 0.1) × 10 22 s −1 can be deduced.Application of the method proposed in Ref. [1] for estimating the uncertainty to the calculation based on the NN interaction from 2017 [4], say, leads to a slightly smaller uncertainty.We have also varied the deuteron wave functions alone.As an extreme case we even took wave functions from phenomenological NN potentials derived in an entirely different framework, namely in the meson-exchange picture [37,38].Also here the obtained values for R remained within the range given above.Finally, omitting the D-wave component of the deuteron wave function in our calculation causes a 5 % variation.It leads to an increase of R and, thus, does not bring the result closer to the values presented by Oosterhof et al.Overall, we confirm the observation by Dover et al. that the predictions for R are fairly insensitive to the details of the employed NN potentials [23], provided that these potentials describe the pp data at low energies.
At this stage we do not have a sensible explanation for the difference of our results (and those of Ref. [23]) to the ones of Oosterhof et al. [29].However, we believe that it is due to the fact that in the latter work the width Γ d is evaluated following the perturbative scheme developed by Kaplan, Savage, and Wise [31].In that scheme there is no proper deuteron wave function.Rather one works with an effectively constructed wave function that is represented in terms of an irreducible two-point function [29,31].This seems to work well for some electromagnetic form factors of the deuteron, at least at low momentum transfer [31,40].On the other hand, the quadrupole moment of the deuteron is overestimated by 40 % [31], which hints that the properties of the wave function at large distances (small momenta) are not that well represented in this scheme.Clearly, this should have an impact on the quantity studied in the present work as well.Note that a comparable agreement (mismatch) with regard to the KSW scheme has been also observed in studies of the electric dipole moment (magnetic quadrupole moment) of the deuteron [41][42][43].In any case, one should not forget that there is convergence problem of the KSW approach for NN partial waves where the tensor force from pion exchange is present [44].It affects specifically the 3 S 1 -3 D 1 channel where difficulties appear already for momenta around 100 MeV/c, see [44] and also the discussions in Refs.[3,45].

Summary
Chiral effective field theory does not only work very well for the interaction between nucleons [3,6] it can be also successfully employed to study the antinucleon-nucleon interaction as has been demonstrated by the Jülich-Bonn group [4,5].In this contribution we have discussed our most refined NN potential [4], established within chiral EFT up to N 3 LO, and we presented some selective predictions for np scattering.The potential has been fitted to up-to-date NN phase shifts and inelasticities provided by a phase-shift analysis of available pp scattering data [7].
As an application of this interaction neutron-antineutron oscillations in the deuteron have been considered.In particular, results for the deuteron lifetime have been presented, evaluated in terms of the free-space n − n oscillation time, utilizing that NN potential in combination with an NN interaction likewise derived within chiral effective field theory.The value obtained for the so-called reduced lifetime R which relates the free-space n − n oscillation time τ n−n with the deuteron lifetime is found to be R = (2.6 ± 0.1) × 10 22 s −1 , where the quoted uncertainty is due to the NN and NN interactions (wave functions).
Our prediction for R agrees with the value obtained by Dover and collaborators almost four decades ago [23] but deviates from recent EFT calculations, based on the perturbative scheme proposed by Kaplan, Savage, and Wise [29], by about a factor of 2. A possible explanation for the difference could be that the KSW scheme does not involve a proper deuteron wave function.Rather this ingredient is represented effectively in terms of an irreducible twopoint function.It is known from past studies that the KSW approach fails to describe quantities that depend more sensitively on the wave function like, for example, the quadrupole moment of the deuteron [31].

Figure 1 . 2 . 4 EPJFigure 3 .
Figure 1.Real and imaginary parts of the phase shifts in the 3 S 1 -3 D 1 partial wave with isospin I = 1.Results at N 3 LO (black solid line), N 2 LO (blue dashed line), and NLO (red dotted line) are shown.The estimated uncertainty is indicated by bands: N 3 LO (magenta), N 2 LO (cyan), and NLO (yellow).The filled circles represent the solution of the pp PWA [7].