PHYSICAL REVIEW D 97, 056008 (2018)

Phenomenology of -antineutron conversion

† Susan Gardner1,2,* and Xinshuai Yan1, 1Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA 2Department of Physics and Astronomy, University of California, Irvine, California 92697, USA

(Received 14 November 2017; published 13 March 2018)

We consider the possibility of neutron-antineutron (n − n¯) conversion, in which the change of a neutron into an antineutron is mediated by an external source, as can occur in a scattering process. We develop the connections between n − n¯ conversion and n − n¯ oscillation, in which a neutron spontaneously transforms into an antineutron, noting that if n − n¯ oscillation occurs in a theory with number minus number (B-L) violation, then n − n¯ conversion can occur also. We show how an experimental limit on n − n¯ conversion could connect concretely to a limit on n − n¯ oscillation, and vice versa, using effective field theory techniques and baryon matrix elements computed in the MIT bag model.

DOI: 10.1103/PhysRevD.97.056008

I. INTRODUCTION no longer the same, so that one cannot convert spontaneously into the other and satisfy energy-momentum Establishing that the symmetry of minus conservation, a constraint that unbroken Lorentz symmetry lepton number (B-L) is broken in nature would demonstrate demands. It is technically possible, but experimentally that dynamics beyond the (SM) exists. This involved, to remove and magnetic fields to the prospect is often discussed in the context of the origin of the extent that sensitive experimental searches with free , with B-L violation necessary both to make become possible. The most recent, and most neutrinoless double β (0νββ) decay possible and to give the sensitive, of such experimental searches was completed neutrino a Majorana mass [1–3]. The would-be mechanism more than 20 years ago [10], yielding a nn¯ lifetime limit of of 0νββ decay is unknown, so that it need not be realized 8 τ ¯ ¼ 0.85 × 10 s at 90% confidence level (C.L.). A through the long-range exchange of a Majorana neutrino, nn next-generation experiment is also under development nor need it even utilize at all—yet its observation [11,12]. Independently, searches for neutron-antineutron would imply the neutrino has a Majorana mass [4]. In this oscillations in nuclei have been conducted, with the most paper we discuss the complementary possibility of B-L stringent lower limit on the bound neutron lifetime being violation in the sector [5,6], and we develop new 1.9 × 1032 yr at 90% C.L. for neutrons in 16O [13]. pathways to its discovery through the consideration of Employing a probabilistic computation of the nuclear n − n¯ conversion. suppression factor [14,15], with realistic nuclear optical We draw a distinction between a n − n¯ oscillation potentials [15], yields an equivalent free neutron lifetime of [5,7,8], in which the neutron would spontaneously trans- 2 7 108 form into an antineutron with the same energy and . × s at 90% C.L. [13]. A recent study of the bound neutron lifetime in [16] also employs a prob- momentum, and a n − n¯ conversion, in which the neutron abilistic framework [17] to determine that the equivalent would transform into an antineutron as mediated by an 1 23 108 external source. The ability to observe n − n¯ oscillations is free neutron lifetime is no less than . × sat famously fragile and can disappear in the presence of 90% C.L. We note that the ability of the free neutron ordinary matter and magnetic fields [6,8,9]. Such environ- experiment to observe a nonzero effect at its claimed mental effects impact the neutron and antineutron differ- sensitivity has recently been called into question [18], ently, in that the energies of a neutron and antineutron are due to the use of a probabilistic, rather than a quantum kinetic, framework for its analysis. We thus find it of particular interest to explore pathways to B-L violation for *[email protected] which these limitations do not apply. † [email protected] In this paper, we consider how it may be possible to observe B-L violation with without requiring that a Published by the American Physical Society under the terms of neutron spontaneously oscillates into an antineutron. One the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to alternate path, that of dinucleon decay in nuclei [19 22],is the author(s) and the published article’s title, journal citation, known and is being actively studied [23–26], though its and DOI. Funded by SCOAP3. sensitivity is limited by the finite density of bound nuclei.

2470-0010=2018=97(5)=056008(17) 056008-1 Published by the American Physical Society SUSAN GARDNER and XINSHUAI YAN PHYS. REV. D 97, 056008 (2018)

Another possibility occurs if the neutron transforms into an n − n¯ oscillations within an effective Hamiltonian frame- antineutron while coupling to an external vector current, as work [6]; we employ its -dependent version [33] to possible in a scattering process. The latter is not sensitive to show how the spin dependence of n − n¯ conversion leaves the presence of matter and magnetic fields, because the the transition probability unsuppressed in the presence of a external current permits energy-momentum conservation to magnetic field. To redress the possibility of suppression be satisfied irrespective of such effects. As we shall see, the from matter effects, however, a different experimental leading-dimension effective operators that realize this are of concept is needed unless the matter were removed—we higher mass dimension than those that give rise to n − n¯ refer to Ref. [18] for a discussion of the implied exper- oscillations, or dinucleon decay; however, the difference in imental requirements. We develop the possibility of n − n¯ mass dimension need not be compensated by a new physics conversion through scattering in Sec. III. Here our par- mass scale—and thus the amplitude for a n − n¯ conversion ticular interest is how it might be connected theoretically to need not be much smaller than that for a n − n¯ oscillation. the possibility of n − n¯ oscillation. We do this by working Moreover, since the neutron is a composite of , and at an energy scale high enough to resolve the quarks in the the quarks carry both electric and color charge, operators ; thus to realize n − n¯ conversion we start with the that mediate n − n¯ oscillations can be related to those that quark-level operators that mediate n − n¯ oscillation and generate n − n¯ conversion. To make our discussion con- dress the quarks with to enable electromagnetic crete, we consider the example of -neutron scatter- scattering. With this we find the quark-level operators that ing, so that n − n¯ conversion would be mediated by the mediate n − n¯ conversion. In Sec. IV we compute the electromagnetic current, though free neutron targets are not matching conditions to the -level effective theory, practicable. Rather, an “effective” neutron target such as 2H computing the needed matrix elements in the MIT bag or 3He would be needed, though the neutron absorption model [34,35]. This gives us a concrete connection between 3 − ¯ cross section on He is too large to make that choice n n conversion and oscillation. Finally in Sec. VI we practicable. The reactions of interest would thus include analyze the efficacy of different experimental pathways to − ¯ e þ 2H → e þ n¯ þ Xðn; pÞ, or, alternatively, either n þ produce n n conversion, and particularly the best indirect 2 n − n¯ e → n¯ þ e or n þ H → n¯ þ p þ XðeÞ, where Xðn; pÞ, limit on oscillation parameters, and we conclude with a summary and outlook in Sec. VII. In a separate paper we e.g., denotes an unspecified final state containing a neutron − ¯ and a . Studies with heavier nuclei, generally develop how best to discover n n conversion in its own right [36]. n þ A → n¯ þ Xðn; pÞ, with A denoting a nucleus such as 58 Ni, could also be possible. n − n The observation of a n − n¯ transition would speak to new II. LOW-ENERGY ¯ TRANSITIONS physics at the TeV scale, and particular model realizations WITH SPIN contain not only TeV-scale new physics but also give At low-energy scales we can regard neutrons and neutrinos suitably sized Majorana [5,27,28]. antineutrons as effectively elementary and realize However, proving that the neutrino has a Majorana mass n − n¯ transitions through B-L violating effective operators in the absence of the observation of 0νββ decay requires not in these degrees of freedom. Previously we have shown that only an observation of B-L violation, but also that of the unimodular phases associated with the discrete sym- another baryon number violating process [29]. Improving metry transformations of Dirac must be restricted the experimental limit on the nonappearance of n − n¯ in the presence of B-L violation; particularly, we have oscillations can also severely constrain particular models found that the phase associated with CPT must be of baryogenesis [30–32]. It could also help shed light on the imaginary [37]. We refer the reader to Appendix A for a mechanism of 0νββ decay in nuclei, which could arise from summary of our definitions and conventions. The notion a short-distance mechanism mediated by TeV scale, B-L that Majorana particles, being their own , have violating new physics or a long-range exchange of a special transformation properties under CPT, CP, and C is Majorana neutrino, with new physics appearing, rather, long known, as are their implications for the interpretation at much higher energy scales, as the supposed mechanism of 0νββ experiments [38,39]. More generally, the existence of 0νββ decay. The continued nonappearance of neutron- of phase constraints associated with the discrete symmetry antineutron oscillations and thus of TeV-scale physics may transformations of Majorana fields had already been noted ultimately speak in favor of a light Majorana neutrino by Feinberg and Weinberg [40], as well as by Carruthers mechanism for 0νββ decay. [41], with these authors determining the phase restrictions In this paper we develop the possibility of n − n¯ associated with the C, CP, T, and TP transformations. conversion in a concrete way. We begin, in Sec. II,by Haxton and Stephenson [42] have also analyzed the phase constructing a low-energy, effective Lagrangian in neutron constraint associated with C for a pseudo-Dirac neutrino and antineutron degrees of freedom with B-L violation, in [43], the case most similar to that of the neutron, though which the hadrons also interact with electromagnetic fields they did not analyze the phase constraints associated with or sources. It has been common to analyze the sensitivity to the other discrete symmetries. Under our CPT phase

056008-2 PHENOMENOLOGY OF NEUTRON-ANTINEUTRON CONVERSION PHYS. REV. D 97, 056008 (2018) constraint, there are two leading mass dimension, CPT where we have assumed that j and B are roughly constant. even and Lorentz invariant, n − n¯ transition operators, If B is nonuniform and depends on the transverse coor- T T namely, O1 ¼ n CnþH:c: and O2 ¼ n Cγ5nþH:c: A third dinates x and y, then ωx and ωy can both be nonzero, ω operator appears if we admit an interaction with an external whereas z will vanishqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the absence of an electric field. μ T μ j O3 n Cγ γ5njμ : 2 2 vector current [44]: ¼ þ H c. Note Introducing ω ¼ ω þ ω , we solve for the eigenval- T ∂ xy x y that a B-L-violating interaction of form in C=n þ H:c: ω 0 vanishes under the use of the equation of motion for a ues and eigenvectors of Eq. (2) (with z ¼ ) assuming free Dirac field. With this in hand, we find that the effec- these quantities are constant to determine the probability tive Lagrangian of n − n¯ conversion, mediated through that a neutron with spin s ¼þ transforms into an anti- electromagnetic interactions, is neutron with s ¼. We find 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 δ δ 2 2 2 2 μν T P →¯ t δ ω tω L ¼ in¯=∂n − Mnn¯ − μ n¯σ nFμν − ðn Cn þ H:c:Þ nþ nþ ¼ 2 2 sin þ 0 cos ð xyÞ eff 2 n 2 δ þ ω0 η 2 T μ 5 δ − ðn Cγ γ njμ þ H:c:Þ; ð1Þ ≈ 2 ω 2 2 sin ðt 0Þ; ð3Þ ω0 where n denotes the neutron field with mass M and   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ , jμ is the current associated with a P 2 ω 2 δ2 ω2 n nþ→n¯− ¼ sin ðt xyÞ cos t þ 0 spin 1/2 particle with electromagnetic charge Qe, noting μν qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  F is the electromagnetic tensor, and δ and η are real ω2   0 2 δ2 ω2 constants, associated with n − n¯ oscillation and conversion, þ 2 2 sin t þ 0 δ þ ω0 respectively. Using Maxwell equations with Heaviside- ν ν ≈ 2 ω Lorentz conventions, we can replace the current j ≡ ψγ¯ ψ sin ðt xyÞ; ð4Þ ν μν with fields via Qej ¼ ∂μF as convenient. We have T where the approximate result reports the probability to neglected the possibility of a n Cγ5n þ H:c: term, P although CPT and Lorentz symmetry permits it [37], leading order in B-L violation. We observe that nþ→n¯þ because it does not contribute to n − n¯ oscillations is quenched by the magnetic field, in that it becomes ω ≪ 1 [33,44,45]. The presence of the external current jμ makes negligibly small unless t 0 , which is not surprising P n − n¯ transitions with a flip of spin possible. To illustrate since the spin is not flipped. In contrast, nþ→n¯− is not suppressed. Although we have illustrated the utility of the the importance of this, we now turn to the computation of μ the n − n¯ transition probability in the presence of a j term in a particular case, our conclusion holds more − ¯ nonuniform magnetic field. generally. In particular, we can probe n n conversion To compute the transition probability in an effective through a scattering process, so that the neutron and Hamiltonian framework with spin degrees of freedom [33], antineutron do not have to have the same energy and we must work out the 4 × 4 mass matrix M associated with momentum. Consequently, we are no longer bound to the Eq. (1). Using the i ∈ jnðp; þÞi, jn¯ðp; þÞi, jnðp; −Þi, context of an oscillation framework, and the quenching jn¯ðp; −Þi basis with p ¼ 0, we compute the matrix ele- problems arising from the presence of either matter or L magnetic fields are completely solved. For future reference ments of the Hamiltonian H associated with eff and define we note that free n − n¯ searches are conducted in the so- the elements of M so that Mij ¼hijHjji/2M. Evaluating M explicitly, we note that matrix elements associated called quasifree limit, so that Eq. (3) can be approximated ij δ2 2 ¯ τ with the η-dependent term are spin dependent. Although by t . Thus a limit on the free nn lifetime nn¯ corresponds δ δ τ−1 − ¯ to a limit on via ¼ nn¯ , so that the limit from the ILL magnetic fields do act to suppress n n oscillations −32 δ L η experiment can be expressed as δ ≤ 5 × 10 GeV at mediated by the term in eff, the behavior of the - dependent term is different. Suppose a magnetic field B is 90% C.L. [10]. present. We choose the spin quantization axis so that S is In what follows we develop how limits from low-energy ˆ scattering experiments that would search for n − n¯ tran- aligned with B and the z axis. Defining ω0 ≡ jμnjjBj and ν μν n − n¯ ω ≡ ηj with Qej ¼ ∂μF , we find sitions can connect concretely to limits from oscil- lation searches. Before so doing, we note that an oscillation 0 1 search after the manner of existing experiments [10] could M þ ω0 δ þ ωz 0 ωx − iωy B C also set a limit on η directly by utilizing nonuniform or B δ þ ω M − ω0 ω − iω 0 C M B z x y C nonstationary electromagnetic fields to generate a nonzero ¼ @ A; 0 ωx þ iωy M − ω0 δ − ωz ωxy [36]. However, we think low-energy scattering experi- ments should offer a more practicable way of realizing ωx þ iωy 0 δ − ωz M þ ω0 n − n¯ conversion, and we make detailed sensitivity esti- ð2Þ mates for these processes in Sec. VI.

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n − n III. ¯ TRANSITION OPERATORS Tα β Tγ δ Tρ σ ðO3Þ ¼½uχ Cdχ ½uχ Cdχ ½dχ Cdχ ðT Þ ; ð9Þ AT THE QUARK LEVEL χ1χ2χ3 1 1 2 2 3 3 a αβγδρσ Considering the n − n¯ transition operators of Eq. (1) although only 14 of these 24 operators are independent, from the viewpoint of simple dimensional analysis, we see because the antisymmetric tensors yield the relation- that the mass dimension of δ,[δ], has ½δ¼1, whereas ships [47] ½η¼−2 since ½jμ¼3. Since ½η/δ¼−3, one might think − ¯ O O O O that n n conversion would be suppressed by an additional ð 1Þχ1LR ¼ð 1Þχ1RL; ð 2;3ÞLRχ3 ¼ð 2;3ÞRLχ3 ; ð10Þ Λ3 Λ factor of NP, where NP is the cutoff mass scale of new physics. This is not necessarily true because of the presence and [48] of other energy scales. To illustrate this explicitly, we need O − O 3 O to develop the form of the n − n¯ conversion operators at the ð 2Þmmn ð 1Þmmn ¼ ð 3Þmmn; ð11Þ quark level. We do this by considering energy scales at which the quark structure of the becomes explicit where m, n ∈ ½L; R. If we also demand that the operators 2 1 but are still well below the nominal scale of new physics, be invariant under SUð ÞL ×Uð ÞY, the electroweak gauge Λ ≲ ≪ Λ QCD E NP. In this way we can realize quark-level symmetry of the SM, then finally only four operators are n − n¯ conversion operators through electromagnetic inter- independent [47,48]. For example, actions, by dressing the quarks of the quark-level n − n¯ P O oscillation operators with photons, since the participating 1 ¼ð 1ÞRRR; ð12Þ quarks also carry . P O The effective Lagrangian for n − n¯ oscillations at the 2 ¼ð 2ÞRRR; ð13Þ QCD scale involves operators with six quark fields, and P Tiα jβ Tγ δ Tρ σ ϵ which thus have an associated coefficient of mass dimen- 3 ¼½qL CqL ½uR CdR½dR CdR ijðTsÞαβγδρσ sion −5. Since these operators are key to our work, we ¼ 2ðO3Þ ; ð14Þ briefly summarize their important ingredients. Based on LRR our earlier discussion of the nucleon-level operators, we P Tiα jβ Tkγ lδ Tρ σ ϵ ϵ expect the quark-level “building blocks” to have the 4 ¼½qL CqL ½qL CqL ½dR CdR ij klðTaÞαβγδρσ Tα β 4 O structure q1χ Cq2χ, where the numerical and Greek indices ¼ ð 3ÞLLR; ð15Þ are flavor and color labels, respectively. We work, too, in a chiral basis with χ ∈ L, R and note that each quark block where the Roman indices label the members of a left- appears as a chiral pair, since operators of mixed chirality handed SU(2) doublet. always vanish. The final n − n¯ operators should be com- The matrix elements of these operators have been patible with the hadrons’ flavor content and also be evaluated in the MIT bag model by Rao and Shrock 3 [47] and, much more recently, in lattice QCD [49,50]. invariant under color symmetry, SUð Þc. There are three ways of forming an SU(3) singlet from a product of six Once we have developed the quark-level n − n¯ conversion 3 operators we, too, use the MIT bag model to evaluate their fundamental representations in SUð Þc. However, in the case of quarks of a single generation, only two color tensors matrix elements. We discuss noteworthy technical aspects can occur [46], namely, of this in Appendix B.

n − n ðTsÞαβγδρσ ¼ ϵραγϵσβδ þ ϵσαγϵρβδ þ ϵρβγϵσαδ þ ϵσβγϵραδ; A. From quark-level operators for ¯ oscillation to n − n¯ conversion ð5Þ Since dimensional analysis shows that the effective operator for n − n¯ conversion would be suppressed with ðTaÞαβγδρσ ¼ ϵραβϵσγδ þ ϵσαβϵργδ ð6Þ respect to that for n − n¯ oscillation by three powers of a with ϵ denoting a totally antisymmetric tensor. We refer to new physics mass scale, we wish to explore the manner in Ref. [46] for a discussion of B-L violating operators with which we can use SM physics to find a more favorable arbitrary generational structure. Working in a chiral basis, relationship. In particular, since the quarks carry electric so that qχ ≡ ð1 þ χγ5Þq/2 and χ ¼ (or, equivalently, charge, we explore the possibility that the external source χ R in the n − n¯ conversion operator is the electromagnetic writing qχ with ¼ L), we note, ultimately, that there are − ¯ current. Of course quarks also carry color charge, but the three types of n n operators [47]: ∂μ a 3 associated current Fμν is not SUð Þc gauge invariant. In Tα β Tγ δ Tρ σ what follows we consider each of the n − n¯ transition ðO1Þχ χ χ ¼½uχ Cuχ ½dχ Cdχ ½dχ Cdχ ðT Þαβγδρσ; ð7Þ 1 2 3 1 1 2 2 3 3 s operators in turn and determine the low-energy effective

Tα β Tγ δ Tρ σ operator that follows from evaluating how its quarks ðO2Þχ χ χ ¼½uχ Cdχ ½uχ Cdχ ½dχ Cdχ ðT Þαβγδρσ; ð8Þ 1 2 3 1 1 2 2 3 3 s interact with a virtual generated by a scattered

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FIG. 1. A neutron-antineutron transition is realized through electron-neutron scattering. The virtual photon emitted from the scattered electron interacts with a general six- n − n¯ oscillation vertex. The particular graphs shown illustrate the two possible ways of O attaching a photon to each of the blocks that appear in the ð 1Þχ1χ2χ3 operator of Eq. (7).  X δ 0 0 ρ charged particle, such as an electron. In any particular, δq u¯ ðp ;sÞ=ϵðkÞð1 þ χ2γ5Þu ðp;sÞ − emi Qρ leading-dimension n − n¯ operator, there are three blocks, 2 02 2 χ p − m and in each block there are two charged particles. When a 2  ρ δ 0 0 virtual photon is attached to these blocks, there are six v¯ ðp;sÞ=ϵðkÞð1 þ χ2γ5Þv ðp ;sÞ − Qδ possible ways that correspond to six different Feynman p2 − m2 diagrams, as shown in Fig. 1. Note that we do not attach a 4 4 × ð2πÞ δð Þðp0 − p − kÞ; ð18Þ photon line to the solid “blob” at the center because, as we shall see, this would yield an effect that would be sup- pressed by higher powers of the new physics mass scale. where ϵμ is the polarization vector of the photon, or, finally, To determine the operator structures that emerge upon including electromagnetic interactions, we first compute X   ρ δ δ Qρ Qδ the matrix element for the process q ðpÞþγðkÞ → q¯ ðp0Þ, − q emi u¯ δðp0;s0Þ=ϵðkÞuρðp;sÞ − 2 p02 − m2 p2 − m2 noting that the superscripts are flavor indices. Working in a χ2   chiral basis, the pertinent terms in the interaction ρ δ 0 0 ρ Q Qδ Hamiltonian are þ χ2u¯ ðp ;sÞ=ϵðkÞγ5u ðp;sÞ þ p02 − m2 p2 − m2 × ð2πÞ4δð4Þðp0 − p − kÞ; ð19Þ δ X X q ρT δ δ ρT ρ ρ H ⊃ ðψ χ Cψ χ þ ψ¯ χ Cψ¯ χ ÞþQρe ψ¯ χ =Aψ χ I 2 1 1 1 1 2 2 χ1 χ2 where we have employed the conventions and relationships X 2 2 ψ¯ δ ψ δ of Appendix A throughout. Since p ¼ p0 , we see the þ Qδe χ3 =A χ3 ; ð16Þ χ3 vector term vanishes if Qρ ¼ Qδ, as we would expect from CPT considerations [37].However,ifQδ ≠ Qρ the final result is nonzero even after summing over χ2. Replacing ρ δ where both q and q¯ have mass m. Computing ϵμðkÞ with kμ we see that the Ward-Takahashi identity is satisfied after summing over χ2. For fixed χ2 the identity   Z  also follows once we sum over the photon-quark contri- X δ δ 0 q 4 ρT δ butions that would yield an electrically neutral initial or hq¯ ðp ÞjT −i d xψ χ Cψ χ 2 1 1 final state, as in the case of n − n¯ transitions. Thus we χ1;χ2  Z Z  extract the effective operator associated with the quark- − 4 ψ¯ ρ ψ ρ − 4 ψ¯ δ ψ δ antiquark-photon vertex as × iQρe d y χ2 =A χ2 iQδe d y χ2 =A χ2

ρ × jq ðpÞγðkÞi; ð17Þ δ m qe δT μ ρ δT μ ρ − ðQρψ −χ Cγ ψ χ − Qδψ χ Cγ ψ −χ Þ; ð20Þ p2 − m2 2 2 2 2 using standard techniques [51], noting T is the time- μ ordering operator and the quarks are treated as free fields, noting that only the Cγ γ5 Lorentz structure would survive we find if ρ ¼ δ. For use in the neutron case we recast this as

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mδ e − q ψ δT γμγ ψ ρ ψ δT γμγ ψ ρ We can now proceed with our explicit construction of the 2 2 ðQρ −χC 5 χ þ Qδ χ C 5 −χÞ; ð21Þ − ¯ O p − m n n conversion operator associated with ð 1Þχ1χ2χ3 in Eq. (7), for which the pertinent Feynman graphs appear in μ so that a sum over χ would yield the Cγ γ5 Lorentz Fig. 1. Using the effective vertex in Eq. (21) for the two- structure that appeared in our neutron-level analysis. quark-field-photon block, we see that the enumerated sets ① ② ③ Since we plan to study the χ dependence of the n − n¯ of graphs, , , and , correspond to the effective vertices conversion operator matrix elements, we may make this −2 δ χ e q m αT μ β αT μ β γT δ replacement without loss of generality. Studying the ½u−χCγ γ5uχ þ uχ Cγ γ5u−χ½dχ Cdχ μ μ 3 2 − 2 2 2 dependence reveals the interplay of the Cγ and Cγ γ5 p m Lorentz structures at the quark level, just as studying the ρT σ × ½dχ3 Cdχ3 ðTsÞαβγδρσ; ð22Þ χ1, χ2, and χ3 dependence of the n − n¯ oscillation matrix elements shows the interplay of C and Cγ5 Lorentz δ þe q m αT β γT μ δ γT μ δ ½uχ Cuχ ½d−χCγ γ5dχ þ dχ Cγ γ5d−χ structures, although only C appears in the neutron-level 3 p2 − m2 1 1 analysis. Since the quark is on its mass shell, we also ρT σ 2 2 2 2 × dχ Cdχ T ; 23 have p ¼ m , so that the explicit factor of 1/ðp − m Þ is ½ 3 3 ð sÞαβγδρσ ð Þ problematic. However, the process we have computed ought not occur because it does not conserve electric and charge. Rather, it may occur within a composite operator δ þe q m αT β γT δ for which there is no change in electric charge, so that the ½uχ Cuχ ½dχ Cdχ 3 2 − 2 1 1 2 2 participating quarks appear as part of a hadron state. Indeed p m ρT μ σ ρT μ σ the phenomenon of confinement in QCD reveals that × ½d−χCγ γ5dχ þ dχ Cγ γ5d−χðTsÞαβγδρσ; ð24Þ quarks are never free, so that p2 − m2 does not vanish in the realistic case. We shall revisit its precise evaluation respectively. Combining these vertices gives the effective − ¯ O once the complete n n conversion operator is in place. operator generated from ð 1Þχ1χ2χ3 , namely,

O˜ χμ −2 αT γμγ β αT γμγ β γT δ ρT σ αT β γT γμγ δ γT γμγ δ ρT σ ð 1Þχ1χ2χ3 ¼½ ½u−χC 5uχ þ uχ C 5u−χ½dχ2 Cdχ2 ½dχ3 Cdχ3 þ½uχ1 Cuχ1 ½d−χC 5dχ þ dχ C 5d−χ½dχ3 Cdχ3 αT β γT δ ρT γμγ σ ρT γμγ σ þ½uχ1 Cuχ1 ½dχ2 Cdχ2 ½d−χC 5dχ þ dχ C 5d−χðTsÞαβγδρσ: ð25Þ

μ 2 2 → 2 Finally, including the current term Qej ðqÞ/q that appears p peff for clarity in later use. We now turn to the through electromagnetic scattering, we have the effective n − n¯ conversion operators associated with the other − ¯ − ¯ O O n n conversion operator n n operators, ð 2Þχ1χ2χ3 and ð 3Þχ1χ2χ3 in Eqs. (8) and (9). Although the block structure of these operators is O ˜ χ em Qejμ ˜ χμ quite different from ð 1Þχ1χ2χ3 , determining the effective ðO1Þχ χ χ ¼ðδ1Þχ χ χ ðO1Þχ χ χ ; ð26Þ 1 2 3 1 2 3 3ðp2 − m2Þ q2 1 2 3 operators is nevertheless straightforward. Employing eff Eq. (21) for the structure of each two-quark-photon block we find for the effective operator generated from where ðδ1Þχ χ χ is the explicit low-energy constant 1 2 3 O O ð 2Þχ1χ2χ3 , associated with the ð 1Þχ1χ2χ3 operator and we replace

O˜ χμ αT γμγ β − 2 αT γμγ β γT δ ρT σ αT β γT γμγ δ − 2 γT γμγ δ ρT σ ð 2Þχ1χ2χ3 ¼½½u−χC 5dχ uχ C 5d−χ½uχ2 Cdχ2 ½dχ3 Cdχ3 þ½uχ1 Cdχ1 ½u−χC 5dχ uχ C 5d−χ½dχ3 Cdχ3 αT β γT δ ρT γμγ σ ρT γμγ σ þ½uχ1 Cdχ1 ½uχ2 Cdχ2 ½d−χC 5dχ þ dχ C 5d−χðTsÞαβγδρσ: ð27Þ

− ¯ O˜ χμ The effective n n conversion operator in this case is then by ðTaÞαβγδρσ in Eq. (27) to yield ð 3Þχ1χ2χ3 and finally O˜ χ 2 − 2 ð 3Þχ1χ2χ3 in analogy to Eq. (28). The quantity p m is ˜ χ em Qejμ ˜ χμ effectively the quark “off-shellness” due to binding effects. ðO2Þχ χ χ ¼ðδ2Þχ χ χ ðO2Þχ χ χ : ð28Þ 1 2 3 1 2 3 3 2 − 2 2 1 2 3 2 2 ðpeff m Þ q We assess this by evaluating E − m , where E is the energy of the ground-state quark, as determined in the MIT O O − ¯ Since ð 3Þχ1χ2χ3 has the same block structure as ð 2Þχ1χ2χ3 , bag model. We have checked that the n n matrix elements we can obtain its effective operator by replacing ðTsÞαβγδρσ of all these effective operators satisfy the Ward-Takahashi

056008-6 PHENOMENOLOGY OF NEUTRON-ANTINEUTRON CONVERSION PHYS. REV. D 97, 056008 (2018) identity. Barring the possibility of vanishing n − n¯ with the three diagrams associated with the u − γ − u¯ hadronic matrix elements, we expect to have two n − n¯ vertex, are shown in Fig. 2. The pertinent terms of the conversion operators for every nonredundant n − n¯ oscil- interaction Hamiltonian can now be written as lation operator. X X We detour briefly to consider a particular model of B-L H ⊃ λ ψ ρT ψ δ Δ ψ¯ ρ ψ ρ I ð χ1 C χ1 χ1 þ H:c:ÞþQρe χ2 =A χ2 breaking, in order to demonstrate that our low-energy, χ1 χ2 X X effective-operator analysis does indeed characterize the δ δ μ þ Qδe ψ¯ χ =Aψ χ þ QΔ e ðΔχ Aμ∂ Δχ þ H:c:Þ; 3 3 χ1 1 1 physics at leading power in the new-physics scale. We χ χ pick a popular model in which n − n¯ oscillations are 3 1 generated through spontaneous breaking of a local B − L ð29Þ symmetry associated with the “partial unification” group 2 ⊗ 2 ⊗ 40 40 where Δχ is a real scalar of mass M. Computing the three SUð ÞL SUð ÞR SUð Þ [5], where SUð Þ breaks to 1 3 1 ρ γ → ¯ δ 0 Δ 0 SUð Þc ×Uð ÞB−L at lower energies. A sample Feynman diagrams for q ðpÞþ ðkÞ q ðp Þ, including ðk Þ as an diagram of a jΔBj¼2 vertex, after Fig. 1 of Ref. [5], along intermediate propagator, we find

 λ X 2 0 1 − χ γ 2 0 1 − χ γ δ 0 0 m þ =k ð 1 5Þ ρ ρ m þ =k ð 1 5Þ δ 0 0 − ei Qρu¯ ðp ;sÞ =ϵðkÞð1 þ χ2γ5Þu ðp;sÞ − Qδv¯ ðp;sÞ =ϵðkÞð1 þ χ2γ5Þv ðp ;sÞ 4 −p k0 2 − m2 −p k0 2 − m2 χ1;χ2 ð þ Þ ð þ Þ  μ 0 μ ϵμ 2 − 2 1 δ 0 0 ρ ðkÞð p p þ k Þ 4 ð4Þ 0 0 þ 2QΔ u¯ ðp ;sÞð1 þ χ1γ5Þu ðp;sÞ ð2πÞ δ ðp þ k − p − kÞ: ð30Þ χ1 ðp − p0Þ2 − M2 k02 − M2

Null results from collider searches for colored, scalar assumption that n − n¯ conversion is mediated by particles imply that M can be no less than Oð500 GeVÞ , the possible conversion operators are [52]. In the low-energy limit, we thus have k02 ≪ M2, determined as long as one starts with a complete set of ðp − p0Þ2 ≪ M2, and indeed k0 → 0. We see that the term six-fermion n − n¯ oscillation operators, irrespective of the in which a photon is radiated from a scalar is completely model from which they arise. negligible in the low-energy limit, and the terms in which k 0 appear are also negligible. Finally we thus recover the result IV. MATCHING FROM THE QUARK O χμ of Eq. (21) and the form of ð 1Þχ1χ2χ2 we have found TO HADRON LEVEL previously, noting λ/M6 ¼ δ/2. Therefore, the particular Working in the quark basis, the effective Lagrangian that model we have considered simply serves as a mechanism to mediates n − n¯ transitions without external sources is generate the needed B-L violating interaction. Under the X L ⊃ 0 δ O n/q ð iÞχ1;χ2;χ3 ð iÞχ1;χ2;χ3 þ H:c:; ð31Þ i;χ1;χ2;χ3

where i ¼ 1, 2, or 3, and the prime denotes sums restricted to yield a nonredundant operator set, as per the discussion in Sec. III. We have already employed this form in determining the n − n¯ conversion operators. By analogy, the effective Lagrangian that mediates n − n¯ conversion is X X Lconv ⊃ 0 η χ O˜ χ n/q ð iÞχ1;χ2;χ3 ð iÞχ1;χ2;χ3 þ H:c:; ð32Þ χ i;χ1;χ2;χ3 FIG. 2. A neutron-antineutron transition is realized through electron-neutron scattering in a particular model of B-L violation. though the precise nature of the restrictions in the sums We consider the case in which the virtual photon interacts with n − n¯ Δ 2 requires further consideration. If the appearance of the neutron through a six-fermion j Bj¼ vertex generated conversion derives from that of n − n¯ oscillation via through the spontaneous breaking of B-L symmetry in the model electromagnetic interactions, then the i, χ1, χ2, χ3 sums of Ref. [5]. A sample Feynman diagram is shown inside the big blue circle, where the dashed line denotes a massive colored are restricted as in Eq. (31). Moreover, the low-energy χ scalar. We represent the effective q − γ − q¯ vertex by a red and constants should not depend on , and the surviving terms gold circular area, which itself is realized by coupling the photon follow from computing the difference of χ ¼þand χ ¼ −. to any of the charged particles that appear. We explicitly show the This in turn implies that only some of the possible n − n¯ diagrams that appear within three dashed red circles. oscillation operators contribute to n − n¯ conversion. It is

056008-7 SUSAN GARDNER and XINSHUAI YAN PHYS. REV. D 97, 056008 (2018) precisely this prospect that makes experimental searches those that appear at the quark level by equating the matrix for n − n¯ oscillations and n − n¯ conversion genuinely elements of the pertinent operators. We have complementary. However, once we have operators of form Z ˜ χ 3 ðconvÞ O χ χ χ ¯ p0 0 L p ð iÞ 1; 2; 3 , broader possibilities follow. If we are agnostic hnð ;sÞj d x eff jnð ;sÞi as to their origin, then the redundancies are those that Z follow from the flavor and color structure of the conversion ¼hn¯ ðp0;s0Þj d3xLðconvÞjn ðp;sÞi; ð34Þ operators themselves. Note that in this case we should also q n/q q replace Q /Q ¼ −2 in Eqs. (25) and (27) by g ,an u d ratio where the states with the “q” subscripts are realized at the unknown parameter. We find, e.g., that relations of the form quark level. Explicitly, then, of Eq. (10) exist, ˜ χ ˜ χ ˜ χ ˜ χ δv¯ðp0;s0ÞCuðp;sÞ ðO1Þχ ¼ðO1Þχ ; ðO2 3Þ χ ¼ðO2 3Þ χ : ð33Þ Z 1LR 1RL ; LR 3 ; RL 3 X 0 ˜ χ ¯ p0 0 3 δ O p O ¼hnqð ;sÞj d x ð iÞχ1;χ2;χ3 ð iÞχ1;χ2;χ3 jnqð ;sÞi Moreover, in the case of ð 1Þχ1;χ2;χ3 , the operator is also χ χ χ symmetric under χ → −χ. Beginning with 48 possible i; 1; 2; 3 operators, we find, finally, that there are 6 independent ð35Þ ˜ χ operators of form ðO1Þχ χ χ , and 12 independent operators 1; 2; 3 and O˜ χ 2 of form ð iÞχ1;χ2;χ3 for each i ¼ , 3, though the gratio 0 0 dependent terms should be separated into new operators ηv¯ðp ;s ÞC=jγ5uðp;sÞ if possible. Z X X ¯ p0 0 3 0 η χ O˜ χ p In what follows, we relate the low-energy constants that ¼hnqð ;s Þj d x ð iÞχ1;χ2;χ3 ð iÞχ1;χ2;χ3 jnqð ;sÞi: appear in this Lagrangian to those in the low-energy χ i;χ1;χ2;χ3 Lagrangian at the nucleon level, in which, due to the ð36Þ low energy scale, we regard the neutron and antineutron as pointlike particles. In particular, noting Eq. (1), we relate Using the connections we have derived in Eq. (26) and in the low-energy constants of this effective Lagrangian to and after Eq. (28) we can rewrite the latter as

Z X em Qejμ 0 μ μ η¯ p0 0 γ p ¯ p0 0 3 δ O˜ R − O˜ L p vð ;sÞC=j 5uð ;sÞ¼ hn ð ;sÞj d x ð Þχ χ χ ½ð Þχ1;χ2;χ3 ð Þχ1;χ2;χ3 jn ð ;sÞi; ð37Þ 3ðp2 − m2Þ q2 q i 1; 2; 3 i i q eff i;χ1;χ2;χ3 so that setting limits on η canalsoconstrainthequark- calculation. In this model, the quarks and antiquarks are level low-energy constants associated with n − n¯ oscil- confined in a static, spherical cavity of radius R by a bag lations. We will determine that the operator matrix pressure B, within which they obey the free-particle Dirac elements associated with i ¼ 1 vanish, so that n − n¯ equation. We only need the ground-state solutions, which we − ¯ s r s r α conversion can only probe some of the n n oscillation denote as uα;0ð Þ [vα;0ð Þ] for a quark [antiquark] of flavor . operators. In the matching relations we have assumed that We present their form and comment on the proper definition the quark-level low-energy constants are evaluated at the s of vα;0 in Appendix B. The quantized quark field is given by matching scale, subsuming evolution effects from the X i† ψ i r i p s r α p s r weak to QCD scales. Note, too, that we assume that “δi” αð Þ¼ ½bαsð nÞuα;nð Þþd sð nÞvα;nð Þ; ð38Þ in Eqs. (35) and (37) are the same irrespective of such n;s i i† effects. Considering the matching relation of Eq. (37),we where i is a color index and bα (dα ) denotes a quark η s s see that for a fixed experimental sensitivity to the limit (antiquark) (creation) operator, for which the on δ will be sharpest if q2 ≃ 0. Thus in evaluating ð iÞχ1χ2χ3 non-null anticommutation relations are the hadron matrix elements we wish to choose p ≃ p0.In i p j† p0 δ δ δ δð3Þ p − p0 the next section we compute the pertinent quark-level fbαsð Þ;bβs0 ð Þg ¼ ss0 ij αβ ð Þ; ð39Þ n − n¯ matrix elements explicitly using the MIT bag i p j† p0 δ δ δ δð3Þ p − p0 model, for which the most convenient choice of kinemat- fdαsð Þ;dβs0 ð Þg ¼ ss0 ij αβ ð Þ: ð40Þ p p0 0 ics is ¼ ¼ . The normalized neutron and antineutron wave functions are given by V. MATRIX-ELEMENT COMPUTATIONS pffiffiffiffiffi ↑ 1 ϵijk i† j† − i† j† k† IN THE MIT BAG MODEL jn i¼ð / 18Þ ðbu↑bd↓ bu↓bd↑Þbd↑j0i; ð41Þ ffiffiffiffiffi Since the MIT bag model is well known [34,35], we only p † † † † † n¯ ↑ 1 ϵijk di dj − di dj dk ; briefly summarize the ingredients that are important to our j i¼ð / 18Þ ð u↑ d↓ u↓ d↑Þ d↑j0i ð42Þ

056008-8 PHENOMENOLOGY OF NEUTRON-ANTINEUTRON CONVERSION PHYS. REV. D 97, 056008 (2018)

↑ ≡ 0 where we use jn i jnqð ; þÞi for spin up (and similarly 1 ¯ χ χ χ − χ χ χ for n) and, following Rao and Shrock [47], we write the B32ð 1; 2; 3Þ¼ 3 B22ð 1; 2; 3Þ particular matrix elements of interest to us as − 4 −χ χ 4χ χ Z i½ 1 2Id þ 1 3Ie; ð53Þ O˜ χμ ≡ ¯ ↑ 3r O˜ χμ ↑ h iiχ1χ2χ3 hn j d ð iÞχ1χ2χ3 jn i 1   B33ðχ1; χ2; χ3Þ¼− B23ðχ1; χ2; χ3Þ − 4ið3χ1χ2I Þ; ð54Þ 6 3 f Nα χμ I χ χ χ : 43 ¼ 4π 2 3 ð iÞ 1 2 3 ð Þ ð Þ pα with χμ Z   We note that ðIiÞχ1χ2χ3 are dimensionless integrals, and we ξα 1 refer the reader to Appendix B for the form of the prefactor 2 2 − ˜2 2 2 − ˜2 Ia ¼ dxx ½j0ðxÞ j1ðxÞ j0ðxÞ 3 j1ðxÞ ; and all other technical details. Picking the z component for Z0 ξ evaluation, we have α 2 2 ˜2 2 ˜2 Ib ¼ dxx ½j0ðxÞ − j1ðxÞj0ðxÞj1ðxÞ; ð55Þ 0 χ3 −2 χ χ χ −χ χ χ ðI1Þχ1χ2χ3 ¼ ðB11ð ; 2; 3ÞþB11ð ; 2; 3ÞÞ Z ξα B12 χ1; χ; χ3 B12 χ1; −χ; χ3 2 2 ˜4 þð ð Þþ ð ÞÞ Ic ¼ dxx j0ðxÞj1ðxÞ; χ χ χ χ χ −χ ; Z0 þðB13ð 1; 2; ÞþB13ð 1; 2; ÞÞ ð44Þ ξ α 2 2 ˜2 2 ˜2 Id ¼ dxx ½j0ðxÞþj1ðxÞ½j0ðxÞ − j1ðxÞ χ3 0 ðI Þχ χ χ ¼ðB 1ðχ; χ2; χ3Þ − 2B 1ð−χ; χ2; χ3ÞÞ   j 1 2 3 j j 1 χ χ χ − 2 χ −χ χ × j2ðxÞþ j˜2ðxÞ ; ð56Þ þðBj2ð 1; ; 3Þ Bj2ð 1; ; 3ÞÞ 0 3 1 χ χ χ χ χ −χ þðBj3ð 1; 2; ÞþBj3ð 1; 2; ÞÞ ð45Þ Z ξ α 2 2 ˜2 2 ˜2 2 Ie ¼ dxx ½j0ðxÞþj1ðxÞj0ðxÞj1ðxÞ; for j ¼ ,3,where 0   Z   ξα 1 4 2 2 2 2 2 2 I ¼ dxx ½j ðxÞþj˜ ðxÞ j ðxÞ − j˜ ðxÞ ; ð57Þ B11 χ1; χ2; χ3 −8i I − χ2χ3I − 8χ2χ3I ; f 0 1 0 1 ð Þ¼ a 3 b c ð46Þ 0 3   ˜ 28 noting j1ðxÞ ≡ ϵ0j1ðxÞ. We have three input parameters, B12ðχ1;χ2;χ3Þ¼−8i −2Ia − χ1χ3Ib − 8χ1χ3Ic ; ð47Þ 3 mu, md,andR, which can be determined by fitting the hadron   spectrum. For definiteness we use the same fits as employed 28 by Rao and Shrock [47],inwhichmu ¼ md. Picking the fit B13ðχ1;χ2;χ3Þ¼−8i −2Ia − χ1χ2Ib − 8χ1χ2Ic ; ð48Þ 3 with nonzero quark mass, so that mu ¼ md ¼ 0.108 GeV −1 6 2 3  and R ¼ 5.59 GeV ,weevaluateNα/ð4πÞ pα ¼ 0.529 × 1 2 −5 6 10 GeV (noting ξα ≃ 2.281), as well as Ia ¼ 0.298, B21ðχ1; χ2; χ3Þ¼−4i − I þ χ2χ3I 2 a 3 b I ¼ 0.0344, I ¼ 0.0106, I ¼ 0.396, I ¼ 0.0557,and  b c d e If ¼ 0.460. Dropping the common factor of i, we evaluate χ3 þ 4ðχ1χ3 − χ1χ2ÞIb þ 4χ2χ3Ic ; ð49Þ ðIiÞχ1χ2χ3 numerically and report the results in Table I.Asa  numerical check, we have also computed the matrix elements 1 2 of the n − n¯ oscillation operators ðO Þχ χ χ , and we repro- χ χ χ −4 − χ χ i 1 2 3 B22ð 1; 2; 3Þ¼ i 2 Ia þ 3 1 3Ib duce Rao and Shrock’s results up to an overall factor of −i  [47]. The pattern of these results, i.e., the sign and relative

þ 4ðχ2χ3 − χ1χ2ÞIb þ 4χ1χ3Ic ; ð50Þ sizes, agree with recent lattice QCD results [50],thoughthe latter are of larger absolute size. We note that the results of   Table I bear out the symmetries of Eq. (33) and the χ → −χ 5 26 ˜ χ χ χ χ −4 χ χ 4χ χ symmetry of ðO1Þχ χ χ . Consequently, upon making the B23ð 1; 2; 3Þ¼ i 2 Ia þ 3 1 2Ib þ 1 2Ic ; ð51Þ 1; 2; 3 matrix element combination ðχ ¼ RÞ − ðχ ¼ LÞ,aswould occur if the conversion operators appear only via electro- and O˜ χ magnetic interactions, the contributions from ð 1Þχ1;χ2;χ3 1 vanish as expected. This underscores the complementarity χ χ χ − χ χ χ − ¯ B31ð 1; 2; 3Þ¼ 3 B21ð 1; 2; 3Þ of n n conversion and oscillation searches. Note, more- over, if we were to restrict our consideration to the − 4 −χ χ 4χ χ i½ 1 2Id þ 2 3Ie; ð52Þ conversion operators that stem from electromagnetism and

056008-9 SUSAN GARDNER and XINSHUAI YAN PHYS. REV. D 97, 056008 (2018)

χ3 − ¯ “ ” TABLE I. Dimensionless matrix elements ðIiÞχ1χ2χ3 of n n conversion operators. The column EM denotes the matrix-element combination of ðχ ¼ RÞ − ðχ ¼ LÞ.

I1 I2 I3

χ1χ2χ3 χ ¼ R χ ¼ L EM χ1χ2χ3 χ ¼ R χ ¼ L EM χ1χ2χ3 χ ¼ R χ ¼ L EM RRR 19.8 19.8 0 RRR −4.95 −4.95 0 RRR 1.80 −8.28 10.1 RRL 17.3 17.3 0 RRL −2.00 −9.02 7.02 RRL −1.07 −8.81 7.74 RLR 17.3 17.3 0 RLR −4.09 −0.586 −3.50 RLR 7.20 6.03 1.17 RLL 6.02 6.02 0 RLL −0.586 −4.09 3.50 RLL 6.03 7.20 −1.17 LRR 6.02 6.02 0 LRR −4.09 −0.586 −3.50 LRR 7.20 6.03 1.17 LRL 17.3 17.3 0 LRL −0.586 −4.09 3.50 LRL 6.03 7.20 −1.17 LLR 17.3 17.3 0 LLR −9.02 −2.00 −7.02 LLR −8.81 −1.07 −7.74 LLL 19.8 19.8 0 LLL −4.95 −4.95 0 LLL −8.28 1.80 −10.1

2 1 − ¯ SUð ÞL × Uð ÞY invariant n n oscillation operators, only off-shellness. Noting Table I, we see that other operators O O the terms from ð 3ÞLRR and ð 3ÞLLR would survive. can also be used as distinct choices, with the associated values of β ranging to roughly 10 times smaller or larger. VI. NUMERICAL ESTIMATES AND Returning to Eq. (37) we see we can recast the connection EXPERIMENTAL PROSPECTS to the n − n¯ oscillation parameters more generally by replacing δ with δ˜, where We now turn to the numerical evaluation of the matching   relations in Eqs. (35), (36), and (37). In the kinematic limit hO3i p p0 0 δ˜ ≡ LLR for which ¼ ¼ , the left-hand sides of Eq. (35) and R3 − L3 z x ðI3ÞLLR ðI3ÞLLR Eqs. (36) and (37) become δδs;s0 and ηðj sδs;s0 þ j δs;−s0 þ X y 0 R3 L3 isj δ − 0 , respectively. As a numerical check, we have δ − s; s Þ × ½ð 2Þχ1;χ2;χ3 ððI2Þχ1;χ2;χ3 ðI2Þχ1;χ2;χ3 Þ verified that our numerical matching procedure does not χ1;χ2;χ3 depend on the particular nonzero component we pick. R3 L3 þðδ3Þχ χ χ ððI3Þχ χ χ − ðI3Þχ χ χ Þ; ð61Þ Generally, a plurality of quark-level operators can contrib- 1; 2; 3 1; 2; 3 1; 2; 3 − ¯ ute to either n n oscillation or conversion; however, if a where we neglect any momentum dependence in the single operator dominates each case, then the experimental nucleon matrix element. limits from the two processes can potentially be compared To study the possible limits on η and hence δ˜,we directly. Choosing μ ¼ z as in the last section and suppos- 0 0 consider the process nðpnÞþlðplÞ → n¯ðpnÞþlðplÞ, ing for illustration that only ðO3Þ operates through LLR where l is a charged lepton, or, more generally, any electromagnetism, we find electrically charged particle. We study the limits on η R3 L3 e m Qejz that would arise if we were to neglect the connection to ðδ3Þ ððI3Þ − ðI3Þ Þ ¼ ηj ; ð58Þ LLR LLR LLR 3 2 − 2 2 z n − n¯ oscillations in a separate paper [36]. To make peff m q numerical estimates of the sensitivity to δ˜, we consider whereas ðδ3Þ hO3i ¼ δ. Combining these relations LLR LLR different possible combinations of beam and target—and we can thus relate a would-be limit on η to one on δ, different energy regimes as well, though we restrict our namely, considerations to center-of-mass energies well below the   δ 2 R3 − L3 nucleon-antinucleon production threshold. In all cases, η Qe m ððI3ÞLLR ðI3ÞLLRÞ ˜ ¼ 2 2 2 ; ð59Þ however, the best limits on δ emerge if we consider q 3 p − m hO3i 2 eff LLR kinematics in which the squared momentum transfer q is minimized. so that with η ≡ βδQe2/q2 we have To evaluate the scattering process, we note that the   R3 L3 lowest-order amplitude in our effective field theory is 1 m ððI3Þ − ðI3Þ Þ β ≡ LLR LLR M η¯ 0 γμ T 0 γ γ5 3 2 − 2 O simply i ¼ ulðplÞð ÞulðplÞvn ðpnÞC μ unðpnÞ,so p m h 3iLLR  eff  that the spin-averaged, absolute-squared amplitude jMj2 is 1 0.108 ð−7.74Þ ¼ GeV−1 3 ð0.365Þ2 2.03 8 2 4 β 2δ˜2 2 Qle j j 0 0 jMj ¼ ½ðpl · p Þðpl · p Þ ≃−0.946 GeV−1: ð60Þ q4 n n 0 0 2 2 þðpn · plÞðpl · pnÞ − 2mlM In evaluating the kinematic factors we replace p2 − m2 eff 2 0 2 0 2 2 2 M p · pl − m p · p ; 62 with Eα − m ¼ pα as an estimate of a quark’s þ ð l Þ lð n nÞ ð Þ

056008-10 PHENOMENOLOGY OF NEUTRON-ANTINEUTRON CONVERSION PHYS. REV. D 97, 056008 (2018) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 2 0 2 where we introduce q ≡ ðpl − plÞ and jM j for the 0 2 0 0 2 2 2MðEe − EeÞþ2me − 2EeEe þ 2jpej Ee − me cosθ ¼ 0: quantity in square brackets for subsequent use. The differ- ential cross section is ð65Þ

The cross section grows as θ−4 as θ approaches zero, so that 1 d3p0 1 d3p0 1 ˜ dσ ¼ l n we can assess its size for fixed δ by first estimating the 2π 3 2 0 2π 3 2 0 F ð Þ El ð Þ En minimum value of θ0. Noting Ref. [53], we determine θ0 in 2 4 4 0 0 two different ways and then choose the larger value for our × jMj ð2πÞ δ ðpl þ p − pl − p Þ; ð63Þ n n cross section estimate. (i) Using the Coulomb interaction between the incoming charged particle and the charged 2 2 2 1/2 where the flux factor F is 4ððpn · plÞ − mlM Þ . In what particles of the deuterium target, we estimate follows we evaluate the cross sections for electron-neutron 2 scattering in various kinematics. We employ beam param- 2α 2meα −7 θ0 ≈ ¼ ¼ 5.45 × 10 rad; ð66Þ eters and target densities as established at existing experi- rajpej jpej ments and facilities, or their planned extensions, to estimate ˜ the limits on δ that can emerge through n − n¯ conversion. where we use jpej¼100 MeV. (ii) Alternatively, by the We begin with the case of an electron beam scattering uncertainty principle, hitting an of ra in transverse from a neutron bound in a deuterium target, e.g., because a size implies the incident momentum is smeared by 1/ra, free neutron target is unavailable. In such a scenario the so that the associated minimum scattering angle is converted n¯ is left in situ, to annihilate with the material 1 α around it. In the alternate case of a neutron beam scattering me −5 θ0 ≈ ¼ ¼ 3.73 × 10 rad; ð67Þ from an atomic target, the converted n¯ emerges with jpejra jpej roughly the same momentum as the incoming beam, so that the location of the annihilation products serves as a where the radius of the hydrogen atom is used for ra. Thus background discriminant. In nuclear stability studies, we see that the two angular estimates are not the same, and experimental backgrounds arise from the interactions of we proceed to estimate the total cross section as per atmospheric neutrinos within the experimental volume, σ producing charged and hadrons, and they have σ ≈ d πθ2 Ω × 0; ð68Þ been studied in detail, noting, e.g., Refs. [13,16]. Neutrinos d θ¼θ0 could also potentially mediate the reaction ν¯n → n¯ν, which would seem to suggest that a n → n¯ process could appear and the larger θ0, so that our total cross section estimate is without breaking B-L symmetry. However, this effect ˜ 2 7 −2 should be negligibly small because it contains the product σ ≈ ½jδj 5.12 × 10 GeV ; ð69Þ of a jΔBj¼2 and a jΔLj¼2 transition. In contrast to ˜ nuclear stability studies, it has proved possible to conduct a where jδj is understood to be in units of GeV. We note that sensitive experimental search for free n − n¯ oscillation such as long as pe ≫ me and θ0 ∝ 1/jpej, the estimate no longer ˜ that no background events that would mimic the signal depends on jpej. To determine the sensitivity to δ, we must appear [10]. compute the event rate dN/dt and finally the expected yield of events. We have A. Electron scattering from a deuterium target dN ¼ Lσ ¼ ϕρ Lσ; ð70Þ We evaluate the cross section for electron-neutron dt T scattering, neglecting the effect of nuclear binding. Integrating over phase space using Eqs. (62) and (63), where the luminosity L is in units of particles/s cm2, ϕ is assuming the neutron is initially at rest, we have the flux in units of particles/s, ρT is the target number density, and L is its length. We turn to the DarkLight   2 experiment operating at the Free-Electron Laser (FEL) dσ e4jβj2δ˜ jp0 j2 e facility of Jefferson National Accelerator Laboratory (JLab) ¼ 2 0 0 dΩ 8π Mjp j jp ejðM þ EeÞ − jpejEe cos θ – e for suitable electron beam parameters [54 56]. The beam 2 jM0j energy that experiment employs is 100 MeV, and its current × 4 ; ð64Þ is 10 mA, for a beam power of 1 MW; it also uses a gaseous q 0 − 0 ; 0 χ pn¼peþpn pe Ee¼ e hydrogen target. In our case we would favor a liquid deuterium target, however, because it is denser, noting, e.g., 0 0 where θ is the angle between p e and pe and Ee ¼ χe is the its use in the experiment of Ref. [57], but target heating may solution to preclude its use under DarkLight conditions. To consider

056008-11 SUSAN GARDNER and XINSHUAI YAN PHYS. REV. D 97, 056008 (2018) this issue further we turn to the Qweak experiment at JLab necessary consequence of the kinematics. We use the [58], which uses a liquid hydrogen target and an electron largest allowed scattering angle in our n − e cross section 1 16 θ ≤ ≈ −4 beam with energy E ¼ . GeV and a beam current of estimate, yielding max me/M 5.44 × 10 rad. In the 180 μA, yielding a beam power in this latter case of case of n − p scattering, the scattering angle is no longer 0.209 MW. Thus for the estimate in our case, we suppose restricted, and the most reasonable choice of angle that we can lower the beam energy to 20 MeV, but keep the depends on the momentum of the neutron beam. For same beam current, so that the beam power will be within neutrons with a kinetic energy of 100 MeV, e.g., we find pffiffiffiffiffiffiffiffiffiffiffiffi the range of the Qweak experiment and a liquid deuterium jpnj¼ 2MTn ≈ 447 MeV, and in this case we adapt our θ ≈ target can be used. The electron beam flux in this case is uncertainty principle estimate of Eq. (67) to write min 0 6 1017 −1 . × s , the number density of liquid deuterium is 1/jpnjra and use that angle in our estimate. For cold neutrons, 22 −3 ρd ¼ 5.1 × 10 cm at 19 K [59], and if we suppose a noting the conditions of the experiment of Ref. [10],we 1 m long target and a running time of 1 yr, the sensitivity to consider a kinetic energy of T ¼ 2 × 10−3 eV, which ˜ n δ for N signal events is corresponds to an average neutron wavelength λ ≈ 6.5 Å p ≈ 1 94 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffi and j nj . keV. In this case, the uncertainty principle N events 1 yr 0.6 × 1017 s−1 1m does not provide a useful restriction on the angle, but low- jδ˜j ≲ 2 × 10−15 1 event t ϕ L energy, forward-scattering experiments with neutrons are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi certainly possible nonetheless. The authors of Ref. [62] note 5.1 × 1022 cm−3 that it should be possible to detect a scattering angle of × ρ GeV: ð71Þ 0.003 rad, and for definiteness we employ this angle for our low-energy n − p cross section estimate. Herewith we summarize our n − e and n − p cross section results. For jp j¼0.447 GeV we have B. Neutron scattering from atomic targets n We now turn to the evaluation of neutron scattering from σ δ˜ 25 74 107 −2; the charged particles of an atomic target. In this case the n−e ¼½j j . × GeV ˜ 2 2 −2 scattering cross section is dominated by the electron σn−p ¼½jδj 3.96 × 10 GeV ; ð72Þ contribution, and we can ignore the role of electromagnetic n − p scattering completely. However, it is important to p 1 94 pick a target for which the loss of neutrons from neutron whereas for j nj¼ . keV we have capture would be minimal. In this regard, a deuterium target would be a good choice because the measured thermal ˜ 2 21 −2 σn−e ¼½jδj 0.881 × 10 GeV ; neutron capture cross section on deuterium is merely σ ¼ c σ δ˜ 21 98 1013 −2 0.508 0.015 mb [60], so that neutron capture effects n−p ¼½j j . × GeV : ð73Þ would have a very limited impact on the transmitted flux. That is, noting that the neutron flux loss would be The n − e and low-energy n − p results should be regarded ϕ ϕ −σ ρ ϕ ϕ ≃ controlled by ¼ 0 expð cL dÞ,wehave / 0 as lower bounds. Nevertheless, it is apparent that the effects of 0.998 if L ¼ 1 m. Although the capture cross section n − p interactions are relatively negligible, and we ignore scales inversely with the neutron velocity at low energies, them in our sensitivity estimates to follow. For the cold capture effects are also negligible for cold neutron energies, neutron case, we employ the beam parameters of the ILL ≈ 10−3 11 −1 noting a kinetic energy of Tn eV for reference. Thus experiment [10], so that we use ϕ ≃ 1.7 × 10 s in our we regard a deuterium target as a suitable choice, though estimate. For the higher energy case, we note the study of 16 there are others, notably one of O. Potentially one could high-energy (1–120 MeV) neutron flux spectra at the also consider neutron scattering from an electron plasma, Spallation Neutron Source (SNS) in Fig. 5.12 of Ref. [63], confined by a magnetic field, but in this case the electron so that we employ a flux of ϕ ¼ 5 × 108 s−1 in that case. — density is limited [61] and much greater electron number Thus for jp j¼0.447 GeV the sensitivity to jδ˜j is densities can easily be found in atomic targets. Thus we n focus on the use of deuterium and oxygen targets. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffi To determine the differential cross section for neutron- N events 1 yr 5 × 108 s−1 1 m jδ˜j ≲ 2 × 10−11 electron scattering, we need only switch the roles of the 1 event t ϕ L neutron and electron in the evaluation of the phase space sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi integrals in Eq. (64). Certainly the differential cross section 5 × 1022 cm−3 is still largest in the forward direction, and we continue to × ρ GeV; ð74Þ use Eq. (68) to estimate the total cross section. Solving 0 the analog of Eq. (65) for En reveals that the neutron 2 2 2 scattering angle is restricted; that is, me − M sin θ ≥ 0 is a whereas for jpnj¼1.94 keV, we have

056008-12 PHENOMENOLOGY OF NEUTRON-ANTINEUTRON CONVERSION PHYS. REV. D 97, 056008 (2018) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffi N events 1 yr 1.7 × 1011 s−1 1 m oscillation, in which a neutron spontaneously converts into jδ˜j ≲ 3 × 10−19 an antineutron, the process is not sensitive to the presence 1 event t ϕ L sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of fields and matter in the external environment because energy-momentum conservation is ensured through the 5 × 1022 cm−3 participation of an external current. We have developed × ρ GeV: ð75Þ the connections between n − n¯ conversion and n − n¯ oscillation, noting, in particular, that operators that give 16 Finally, we turn to the case of a solid O target, for which the rise to spontaneous n − n¯ transitions can also give rise to 22 −3 density at 24 K is ρo ¼ 5.76 × 10 cm [64]. Here, too, we n − n¯ conversion via an external electromagnetic current, focus on the n − e scattering contribution. Since each O atom because the quarks carry electric charge. We have deter- has eight , the cross section should be 8 times larger. mined precisely how quark-level conversion operators can Thus for jpnj¼0.447 GeV we estimate a sensitivity of be determined in this case and, moreover, how only certain − ¯ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffi of the operators that generate n n oscillation can also 1 5 108 −1 1 generate n − n¯ conversion. Thus searches for n − n¯ con- ˜ −12 N events yr × s m jδj ≲ 7 × 10 version are genuinely complementary to those for n − n¯ 1 event t ϕ L sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oscillation. 5.76 × 1022 cm−3 We have also studied the inferred limits on the subset of − ¯ × ρ GeV; ð76Þ low-energy constants associated with n n oscillation that could arise from n − n¯ conversion searches. We have found p 1 947 that the connection is sharpest when the momentum whereas for j nj¼ . keV we have transfer associated with the scattering is smallest, so that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffi the higher-mass-dimension conversion operator is the least N events 1 yr 1.7 × 1011s−1 1 m suppressed. We have, moreover, evaluated a number of jδ˜j ≲ 1 × 10−19 1 event t ϕ L electron-neutron scattering processes and have found that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cold neutron beams scattering from atomic (or molecular) 5.76 × 1022 cm−3 deuterium or oxygen targets appear to have the greatest × ρ GeV: ð77Þ sensitivity. Generally our anticipated limits are much less severe than those associated with direct searches for n − n¯ It appears that the greatest sensitivity to the n − n¯ oscillation oscillation, recalling that the free neutron limit from the ILL δ ≤ 5 10−32 parameter δ˜ can be realized through cold neutron beams experiment can be expressed as × GeV at scattering from atomic (or molecular) deuterium or 16O 90% C.L. [10], but the set of probed operators is different. targets. We wish to emphasize that these particular estimates Also a quantitative understanding of the manner in which the spontaneous process is suppressed by external fields rely on choosing the largest value of a very small scattering and matter is necessary for assessing those limits. The study angle, supposing that the detection of annihilation events of B-L violation in scattering does not have such a liability, would rely on their displacement away from the forward and we note the prospects for the discovery of B-L violation direction. We note that the measurement of much smaller via n − n¯ conversion without reference to n − n¯ oscillation momentum transfers in neutron scattering than we have in a separate paper [36]. considered are under development [65], and the realization Finally we would like to note that the mechanism of of this could ultimately lead to significant improvements in n − n¯ conversion can lead to broader studies of the spin and sensitivity. Improvements could also come from the use of flavor dependence of B-L violating processes. We note the brighter neutron beams. This could be realizable, e.g., with prospect of Δ0 − Δ¯ 0 transitions, as well as that of n − Δ¯ 0, the planned LD2 cold source for the NG-C guide at NIST, although the quark-level operators that appear are shared by where we refer to Fig. 8 in Ref. [66] for further details. The n − n¯ conversion. It is also possible to probe B-L violating best prospects in this regard, however, should be offered by operators that also change strangeness, mediating, e.g., the European Spallation Source (ESS), noting Refs. [11,12] ¯ ¯ 0 ¯ n − Λ and n − Σ transitions [21],orΛ0 − Λ0 transitions for a description of the possibilities. [67,68]. More generally, we note that the ongoing technical efforts in the realization of the next generation of high- VII. SUMMARY AND OUTLOOK intensity electron and neutron beams can have an imme- diate impact on fundamental physics through searches for In this paper we have considered the process of n − n¯ B-L violation via n − n¯ conversion. conversion, in which a jΔBj¼2 transition, which breaks B-L symmetry, is mediated by an external source. The ACKNOWLEDGMENTS observation of such a process would reveal the existence of physics beyond the SM and indeed that of fundamental We acknowledge partial support from the U.S. Majorana dynamics. In contradistinction to n − n¯ Department of Energy Office of under

056008-13 SUSAN GARDNER and XINSHUAI YAN PHYS. REV. D 97, 056008 (2018) pffiffiffiffiffiffiffiffi 3/2 † Contract No. DE-FG02-96ER40989, and we thank our jnðp;sÞi ¼ 2Epð2πÞ b ðp;sÞj0i: ðA6Þ colleagues, particularly Jonathan Feng, at the University of California, Irvine (UCI), for generous hospitality. X. Y. With these choices we recover the usual form of the equal- would also like to thank the Graduate School of the time commutation relations in ψðxÞ and ψ¯ ðxÞ and that of University of Kentucky and the Huffaker Fund for provid- hnðp;sÞjnðp0;s0Þi [51]. ing travel support to UCI. S. G. is also grateful to the Mainz We also note the convenient relationships for Institute Theoretical Physics for partial support and gracious hospitality during the final phases of this project. vTðp;sÞC ¼ u¯ðp;sÞ; uTðp;sÞC ¼ v¯ðp;sÞ; ðA7Þ We also thank Robert Jaffe for key correspondence regarding antiquark states in the MIT bag model and as well as Christopher Crawford, Geoffrey Greene, Wick Haxton, Shannon Hoogerheide, Wolfgang Korsch, Kent Leung, u¯ðp;sÞγμuðp0;s0Þ¼v¯ðp0;s0Þγμvðp;sÞ; ðA8Þ Bradley Plaster, and Michael Snow for helpful comments μ 0 0 0 0 μ and/or references. u¯ðp;sÞγ γ5uðp ;sÞ¼−v¯ðp ;sÞγ γ5vðp;sÞ; ðA9Þ

APPENDIX A: DEFINITIONS AND where the latter follow from computing the transpose of the CONVENTIONS left-hand side in each case. In this appendix we collect the definitions and basic results that underlie the central arguments of the paper. APPENDIX B: THE GROUND-STATE The discrete-symmetry transformations of a four-component IN THE MIT BAG MODEL fermion field ψðxÞ are given by In the MIT bag model, the quarks and antiquarks obey the free-particle Dirac equation within a spherical cavity −1 0 2 c CψðxÞC ¼ ηcCγ ψ ðxÞ≡ηciγ ψ ðxÞ≡ηcψ ðxÞ; ðA1Þ of radius R, subject to boundary conditions at its surface [34,35]. Solutions of opposite parity, i.e., with k ¼ ∓1 −1 0 Pψðt; xÞP ¼ ηpγ ψðt; −xÞ; ðA2Þ exist,   −1 1 3 ðsÞ Tψðt; xÞT ¼ ηtγ γ ψð−t; xÞ; ðA3Þ Nα ij0ðpαrÞχ ψ α −1 ðrÞ¼pffiffiffiffiffiffi ; ðB1Þ ðk¼ Þ ðsÞ 4π −ϵαj1ðpαrÞσ · rˆχ where η , η ,andη are unimodular phase factors of the c p t   charge-conjugation C,parityP, and time-reversal T trans- ˜ ðsÞ Nα ij1ðpαrÞσ · rˆχ formations, respectively, and we have chosen the Dirac-Pauli ψ α 1 ðrÞ¼pffiffiffiffiffiffi ; ðB2Þ ðk¼ Þ 4π ϵ r χðsÞ representation for the gamma matrices. Furthermore the αj0ðpα Þ unimodular factors are constrained so that η η η and η are c p t p where pure imaginary [37].   The plane-wave expansion of a Dirac field ψðxÞ (noting 2 −2 1 ξ ξ 2 ℏ 1 αj0 ð αÞ ¼ c ¼ ) is given by Nα ; ¼ 3 2 − 1 ðB3Þ Z R ½ EαRðEαR ÞþmR 3p X   d −ip·x 2 −2 1 ψðxÞ¼ pffiffiffiffiffiffi fbðp;sÞuðp;sÞe ξ ξ 2 2π 3/2 2 ˜ αj1 ð αÞ ð Þ E s¼ Nα ¼ 3 ; ðB4Þ R ½2EαRðEαR þ 1ÞþmR þ d†ðp;sÞvðp;sÞeip·xg; ðA4Þ and jn is a spherical Bessel function of the first kind of with spinors defined as order n, with m and s denoting the quark mass and spin, respectively. Also  s   σ·p 0ðsÞ  χð Þ χ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p;s N ; v p;s N EpþM ; ð Þ¼ σ·p χðsÞ ð Þ¼ ξ − E M χ0ðsÞ α 2 2 2 Eα m pþ pα ¼ ;Eα ¼ pα þm ; ϵα ¼ ; ðB5Þ R Eα þm ðA5Þ ξ 0ðsÞ 2 ðsÞ þ 1 − 0 where an eigenvalue equation determines the quantity α noting χ ¼ −iσ χ , χ ¼ð0Þ, χ ¼ð1Þ, and N ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and eventually the energy Eα, namely, Ep þ M. The pertinent fermion anticommutation p † p0 0 p † p0 0 relations are fbð ;sÞ;b ð ;sÞg ¼ fdð ;sÞ;d ð ;sÞg ¼ j1ðξαÞ¼ϵαj0ðξαÞ; ðB6Þ δð3Þðp − p0Þδss0 ; all others vanish. We also give the single-particle states a covariant normalization; e.g., or, equivalently,

056008-14 PHENOMENOLOGY OF NEUTRON-ANTINEUTRON CONVERSION PHYS. REV. D 97, 056008 (2018)

FIG. 3. Graphical illustration of the solutions to the eigenvalue equation of Eq. (B6) for a ground-state (a) quark, which has k ¼ −1, and (b) antiquark, which has k ¼ 1. Note that the solid curve denotes j0 in each case, whereas the dashed line is j1/ϵα in (a) and −j1/ϵα in (b). The solid red dot shows the proper solution in each case.

  kξα ¯ − r σ rˆχ0ðsÞ ξ Nα ij1ðpα Þ · tan α ¼ ðB7Þ vα 0ðrÞ¼pffiffiffiffiffiffi ; ðB10Þ k − kmR þ EαR ; 0ðsÞ 4π ϵ¯αj0ðpαrÞχ for k −1 or k 1 − . The prescription for a ground- ¼ ðþÞ ¼ ð Þ where state antiquark has not been clearly stated [69], and   mistakes have appeared in the literature [70]. Here we 2 −2 1 ξ ξ 2 ¯ αj1 ð αÞ clarify the proper choice through Fig. 3, which illustrates Nα ¼ 3 ∓1 R ½−2EαRð−EαR þ 1ÞþmR the solutions to the eigenvalue equation for k ¼ . The   2 −2 1 ground-state solution for a quark is given by the solid red ξ ξ 2 αj0 ð αÞ ϵ¯−1 dot in Fig. 3(a). In order for the magnitude of the energy of ¼ 3 α ; ðB11Þ R ½2EαRðEαR − 1ÞþmR the ground-state antiquark to be the same as that of the antiquark we must also pick the solution marked by the with solid red dot in Fig. 3(b). Consequently the quark and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi antiquark solutions are related by −Eα − m −1 ϵ¯α ¼ ¼ ϵα : ðB12Þ ¯ − ξα ¼ −ξα; ðB8Þ Eα þ m ¯ Eα ¼ −Eα; ðB9Þ Putting everything together we have   ¯ ¯ ϵ r σ rˆχ0ðsÞ where ξα and Eα denote those of the ground-state anti- s Nα i αj1ðpα Þ · vα 0ðrÞ¼pffiffiffiffiffiffi : ðB13Þ s r ψ r ; 0ðsÞ particle. The quark state is uα;0ð Þ¼ αðk¼−1Þð Þ. In con- 4π −j0ðpαrÞχ s r trast, the true solution for an antiquark vα;0ð Þ should be

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