Gauged B-L Number and Neutron--Antineutron Oscillation: Long-range Forces Mediated by Baryophotons

Transformation of neutron to antineutron is a small effect that has not yet been experimentally observed. %\cite{Phillips:2014fgb}. In principle, it can occur with free neutrons in the vacuum or with bound neutrons inside the nuclear environment different for neutrons and antineutrons and for that reason in the latter case it is heavily suppressed. Free neutron transformation also can be suppressed if environmental vector field exists destinguishing neutron from antineutron. We consider here the case of a vector field coupled to $B-L$ charge of the particles ($B-L$ photons) and study a possibility of this to lead to the observable suppression of neutron to antineutron transformation. The suppression effect however can be removed by applying external magnetic field. If the neutron--antineutron oscillation will be discovered in free neutron oscillation experiments, this will imply limits on $B-L$ photon coupling constant and interaction radius few order of magnitudes stronger than present limits form the tests of the equivalence principle. If $n-\bar n$ oscillation will be discovered via nuclear instability, but not in free neutron oscillations in corresponding level, this would indicate to the presence of fifth-forces mediated by such baryophotons.


Introduction
The oscillation phenomenon between the neutron and antineutron, n →n was suggested in early 70's by Kuzmin [1]. First theoretical scheme for n −n oscillation was suggested in Ref. [2], followed by other models as e.g. [3,4,5]. Experimental observation of the transformation of neutron to antineutron n →n would be a demonstration of baryon number violation by two units, from B = +1 for neutron to B = −1 for antineutron. This will be an experimental demonstration that one of the Sakharov's conditions [6] required for the generation of baryon asymmetry in the universe is indeed realized in the nature. The n →n conversion so far was not experimentally observed. However, this does not exclude the possibility that it can be a rare/suppressed process. For a review of the present theoretical and experimental situation on n −n oscillation, see [7].
Apart of baryon-conserving (large) Dirac mass term m n nn, the neutron may acquire a small Majorana mass term, ε nn (nCn + h.c.), which violates the baryon number by two units and induces the neutron -antineutron mass mixing. As far as the neutron is a composite particle, n −n mixing can be induced by the effective six-fermion operators involving the first family quarks u and d: where M is some large mass scale of new physics beyond the Standard Model. These operators can have different convolutions of the Lorentz, color and weak isospin indices which are not specified. (More generally, having in mind that all quark families can be involved, such operators can induce the mixing phenomena also for other neutral baryons, e.g. between the hyperon Λ into the anti-hyperonΛ.) The models of Refs. [2,3,4,5] are just different field-theoretical realizations for the operators (1). Taking matrix elements from these operators between the neutron and antineutron states, one can estimate the neutron Majorana mass modulo the Clebsch factors as The coefficients of matrix elements n|uddudd|n for different Lorentz and color structures of operators (1) were studied in ref. [9], but we do not concentrate here on these particularities and take them as O(1) factors.
Concerning the experimental limits on n −n oscillation time τ nn = 1/ε nn , the direct limit on free neutron oscillations imply τ nn > 0.86 × 10 8 s [17]. The nuclear stability limits, with uncertainties in the evaluation of nuclear matrix elements, translate into τ nn > 1.3 × 10 8 s [18] and τ nn > 2.7 × 10 8 s [19]. The latter implies the strongest upper limit on n −n mixing, ε nn < 2.5 × 10 −24 eV. The future long-baseline direct experiment at the European Spallation Source (ESS) can reach the sensitivity down to 10 −25 eV and thus improve the existing limits on n −n oscillation time by more than an order of magnitude [7].
One can consider a situation when baryon number B is broken not explicitly but spontaneously. Such a baryon symmetry can be global or local, with different physical implications.
The possibility of spontaneous violation of global lepton symmetry after which the neutrinos can get non-zero Majorana masses is widely discussed in the literature. As a result, a Goldstone boson should appear in the particle spectrum, named as Majoron [10]. Spontaneous violation of global baryon number in connection with the Majorana mass of the neutron was first discussed in ref. [11], in the context of Mohapatra-Marshak model for n −n oscillation [2]. Recently the discussion was revived by one of us in Ref. [12], where also seesaw model for n−n transition with low scale spontaneous violation of baryon number were suggested. An associated Goldstone particle -baryo-majoron, can have observable effects in neutron to antineutron transitions in nuclei or dense nuclear matter. The low-scale baryo-majoron model [12] has many analogies with the low scale Majoron model for the neutrino masses [13]. By extending baryon number to B − L symmetry, baryo-majoron can be identified with the ordinary majoron associated with the spontaneous breaking of lepton number, with interesting implications for neutrinoless 2β decay with the majoron emission [14], for matter-induced effects of the neutrino decay [15] and for the Majoron field effects in the early Universe [16].
In this paper we discuss a situation when baryon number is related to a local gauge symmetry. The idea to describe the conservation of baryon number B and lepton number L similar to the conservation of electric charge by introducing gauge symmetries U (1) B and U (1) L , i.e. in terms of baryon or lepton charges coupled to the massless vector fields of leptonic or baryonic photons with tiny coupling constants, was suggested long time ago [20]. Their effects for the neutron oscillations were studied in Refs. [21]. Nowadays the limits on such interactions are very stringent. Best limits on the coupling strength of baryonic and leptonic photons were obtained from the Eötvös type of experiments testing the equivalence principle [23]. Then, the common sense argument used here was that coupling of such photons is many order of magnitude weaker than the gravitational interaction between baryons or leptons and therefore such photons are likely non-existent. We will try to revise this concept.
Since baryon number B and lepton number L separately are not conserved due to nonperturbative effects, it is difficult to promote them as gauge symmetries without altering the particle content of the Standard Model, i.e. without introducing new exotic particles. Therefore, we discuss not baryonic and leptonic photons separately but the vector fields associated with U (1) B−L gauge symmetry. In the Standard Model this symmetry is anomaly free and B − L current is conserved at the perturbative as well as at non-perturbative level. On the other side, the existence of neutron-antineutron oscillation or other similar phenomena would imply that this gauge symmetry, if it exists, should be spontaneously broken. n −n mixing cannot be induced without violating B and thus B − L, which should render massive also B − L baryophoton. However, if its gauge coupling constant is very small, such a baryophoton can remain extremely light, and mediate observable long range forces (fifth force) between material bodies.
Clearly, such B − L baryophotons couple with opposite charges not only between the baryons and anti-baryons (and between the leptons and anti-leptons), but also between the baryons and leptons. Thus, B − L charge of the neutral hydrogen atom is zero while the B − L charge of heavier neutral atoms is determined by the number of neutrons in nuclei. Therefore, the regular matter built by nuclei heavier than hydrogen is B − L charged. In principle, at the scale of the universe B − L charge might be compensated by the relic neutrino component that so far remains experimentally undetected.

Experimental limits on B-L photons
The baryophoton b µ associated with U (1) B−L gauge symmetry interacts with the fermion (neutron, proton, electron and neutrino) currents as gb µ (nγ µ n + pγ µ p − eγ µ e − νγ µ ν). As far as the existence of the neutron-antineutron mixing implies the violation of U (1) B−L , this gauge boson cannot remain exactly massless.
In particular, since D = 9 effective operators (1) are now forbidden by U (1) B−L symmetry, they can be replaced by the effective D = 10 operators involving the complex scalar field χ bearing two units of B−L number, Q χ = −2. Its vacuum expectation value (VEV) χ = υ χ . spontaneously breaks the U (1) B−L . By substituting χ → χ in (1), operator (3) reduces to (1) with M 5 = M 6 /υ χ , and thus the induced n −n mixing mass can be estimated as Therefore, taking that the scale M larger than 10 TeV, the neutron-antineutron oscillation can be within the experimental reach at the ESS, i.e. ε nn > 10 −25 eV, if υ χ > 1 keV or so. We take this as a benchmark value for the B − L symmetry breaking scale. If the scale M is taken ad extremis as small as M ∼ 1 TeV, then one would get υ χ ∼ 1 meV. However, such a tiny scale does not seem to be a really realistic, However, a huge hierarchy problem between the scale υ χ ∼ 1 meV and the electroweak scale ∼ 100 GeV will be a headache. More important, it is very very unlikely that the violation of B − L at such a small υ, which is just about a 3 K (cosmic microwave background (CMB) temperature today), can be relevant for primordial baryogenesis. The possibility of very small υ is excluded in realistic models which we discuss later. 1 In principle, also the VEVs v i of other scalars η i with non-zero B − L charges Q i can also participate in breaking U (1) B−L . As a result, the baryophoton should acquire the mass: where g is gauge coupling constant of baryophotons, Thus, the value of υ χ defines the minimal possible value of M b for a given constant g. If there are other scalar fields η i with non-zero Q = B − L charges and non-zero VEVs, their contribution would make M b larger. Therefore, baryophotons should mediate an Yukawa-like fifth-force between the material bodies. Vector boson b µ exchange induces a spin-independent potential energy of the interaction between the test particle with B −L charge Y i , in our case the neutron or antineutron, and an attractor (a massive body as e.g. Earth or sun) with the overall B − L charge Y A : where α B−L = g 2 /4π, in addition to the gravitational potential energy V gr i = −Gm i M A /r, G being the Newton constant. The overall B − L charge of the gravitating body of mass M A is defined by its chemical composition: Q A Y n M A /m n , m n being the neutron mass. Due to electric neutrality, the amount of protons and electrons should be equal and thus their contributions cancel each other. Hence, the value of Q A is determined by the neutron fraction Y n . In particular, Q = 0 for hydrogen and Q ≈ 0.5 for a typical heavy nuclei.
The maximal possible range of the Yukawa radius λ for a given constant g is limited by the minimal value of the symmetry breaking scale, υ = υ χ . If there are other scalar fields η i with different Q = B − L charges and non-zero VEVs, their contribution would make the mass M b larger, and thus would shorten the range of λ. The results of torsion-balance tests of the weak equivalence principle from Ref. [23] can be interpreted as limits on fifth  forces, and in particular, for the force mediated by the B − L baryophotons, as limits on dimensionless constant α B−L for a given radius λ, as shown in Fig. 1.
In difference from universal gravity, the baryophoton exchange would induce for the neutron (Q n = 1) and antineutron (Qn = −1) the potential energies of different sign, Vn = −V n . It is convenient to relate the values V n,n with the neutron (and antineutron) gravitational potential energy V gr = −Gm n M A /r: introducing a dimensionless parameterα = α B−L /Gm 2 n , and q A = Q A /(M A /m n ) being the massive objects B − L charge per neutron mass unit. The upper limits on the parameterα as a function of the radius λ are given in Fig. 6 of Ref. [23] (in Ref. [23] these values are normalized per atomic mass unit, 1 amu= 0.99 m n ). In Fig. 1 these limits of Ref. [23] are shown directly translated for α B−L . As we see, if the Yukawa radius is larger than the Earth diameter, λ > 10 9 cm or so, then the upper limit on α B−L becomes practically independent on λ and it corresponds to α B−L < 10 −49 , orα < 1.7 × 10 −11 [23]. Now we can estimate the neutron potential energy V n as produced due to the baryophoton potentials by the Earth, sun and the Galaxy, relative to the corresponding gravitational potential energies.
The Earth induces the gravitational potential energy for the neutron at its surface V gr E = m n φ gr = −Gm n M ⊕ /R ⊕ ≈ 0.66 eV. The sun gives a bigger contribution, V gr S = −Gm n M /AU ≈ 10 eV. Finally, the Galaxy itself induces even bigger value V gr G ∼ 1 keV. 2 Since the Earth is built by heavy nuclei, B − L charge of the Earth is approximated as 50% of the number of baryons in the Earth, i.e. q E 0.5. The sun is dominantly consists of hydrogen which has vanishing B − L, and thus its fifth force is essentially determined by the mass fraction of heavier nuclei (helium, etc.) with Y n 0.5. Therefore, q S 0.13 as one can estimate from the known chemical composition of Sun [24]. The same applies to the Milky Way contribution, q G 0.13.
Thus, assuming that λ is larger than the Earth diameter, λ > 2R ⊕ , the values of V n at the surface of Earth can be estimated as: 3 From (6), taking α B−L = 10 −49 , i.e.α = 1.7 × 10 −11 , we see that for our benchmark value υ χ = 1 keV we obtain λ ∼ 10 16 cm which is much larger than the sun-Earth distance (1 AU≈ 1.5 × 10 13 cm). Therefore, in the case the contributions of the Earth and the sun in V n can be as large as respectively 0.56 × 10 −11 eV and 2.2 × 10 −11 eV, amounting in total as 2.8 × 10 −11 eV. For λ's smaller than the Earth diameter, the larger values of α B−L are allowed (see Fig. 1) but the available volume of the source drops as (λ/R ⊕ ) 3 and thus upper limit on V n sharply decreases.
It is interesting to question to how large values of λ and how large potential V n can be induced by the Galaxy. For the benchmark value υ χ = 1 keV, we see from (6) that for having λ > 20 kpc = 6 × 10 22 cm, one has to take α B−L < 10 −62 or so, in which case the baryophoton induced potential will be less than 10 −21 eV and therefore it would have no influence for the experimental search of n −n oscillations. Even taking the value of the VEV as small as υ χ = 1 meV, i.e. at its extreme dictated by the value ε nn ∼ 10 −24 eV by the operators (3) with M = 1 TeV, we obtain that λ > 10 kpc = 3 × 10 22 cm can be obtained if α B−L < 6 × 10 −51 or so. In this marginal situation, the Galaxy contribution in V n could amount up to 10 −10 eV. In any case, contribution of more distant objects as neighboring galaxies and galaxy clusters are exponentially suppressed since a very small B − L breaking scale, υ χ < 1 meV, is not of interest.

n −n oscillation in the presence of B −L fifth force
Non-relativistic Hamiltonian that describes n−n oscillation in the presence of fifth force and magnetic fields can be presented as 4 × 4 matrix acting on the state vector (n + , n − ,n + ,n − ) 2 Notice that we are dealing with the gravitational potentials which fall as ∝ 1/r and not with gravitational forces testable by torsion balance experiments. The latter are ∝ 1/r 2 and their hierarchy between the Earth, sun and the Galaxy becomes reordered in opposite way. This is the reason why for λ exceeding the Earth Diameter, the experimental limits of Ref. [23] become independent on λ. 3 We neglect the annual modulation of V gr S due to small variation the sun-Earth distance, as well as potentials induced by other planets and the Moon. The latter also could be responsible for time variation of the total potential. We also neglect contributions from neighboring galaxies and galaxy clusters since describing the neutron and antineutron states with two spin polarizations: 4 where µ n = −6×10 −12 eV/G is the magnetic moment of the neutron, B is the magnetic field and σ 3 is the third Pauli matrix since the spin quantization axis is chosen as the direction of the magnetic field. In this basis one has no spin precession and the Hamiltonian (9) is diagonal. Omitting the universal terms and taking V = V n = −Vn, it can be rewritten as where Ω B = |µ n B| = 6·10 −12 (B/1 G) eV is the Zeeman energy shift induced by the magnetic field. In general case, with V n and Ω B both non-zero, the n andn oscillation probabilities are different between the + and − polarization states: where t is a neutron free flight time. In the realistic experimental conditions t cannot be very large, e.g. it was ∼ 0.1 s in the experiment [17], it can be up to ∼ 1 s in the experimental setup for cold neutrons at the ESS, and in principle it could reach ∼ 10 s in the experiments when the neutrons vertically fall down in a deep mine. Let us discuss first the case when the fifth force is absent, V = 0, and there remains only the magnetic field contribution, i.e. ∆ 2 ± = 4Ω 2 B ε 2 nn . Then the neutron oscillation probabilities of + and − polarization states should be equal and for making effective the oscillation during a time t, the magnetic field should be suppressed to a needed degree. Namely, If Ω B t 1, the oscillations should be averaged in time and one gets P ± nn = ε 2 nn /2Ω 2 B . However, for small free flight times, t < 1 s, the magnetic field can be suppressed achieving Ω B t < 1. The argument of sine wave is small and oscillation probability P (t) becomes practically independent on Ω B : The latter condition is known as "quasi-free" condition. The needed level of magnetic field suppression depends on the neutron free flight time in experimental conditions. For t = 0.1 s, the condition Ω B < t −1 implies Ω B < 10 −15 eV, and thus B < 10 −4 G. Suppressing fields to the level of 1 nT would be sufficient for future realistic experimental times order 1 s. Figure 2: Potential energy V n of the neutron in B − L field of Sun and Earth. Region of potentils V n above the red curve is excluded by torsion balance experiment [23].
Let us consider now the case with non-zero V n . From (6), taking α B−L = 10 −49 , we see that for our benchmark value υ χ = 1 keV we obtain λ ∼ 10 16 cm which is much larger than the sun-Earth distance (1 AU= 1.5 × 10 13 cm). We see from Fig. 1, that for λ > 1 AU, V n can reach the values up to 3 × 10 −11 eV, equivalent to Ω B of the magnetic field B 5 G. This would lead to strong suppression of n −n oscillation even if the magnetic field value vanishing: the n −n oscillation will not be discovered at the ESS even if ε nn > 10 −24 eV. Therefore, for achieving the quasi free condition for n −n oscillation allowing to discover n −n conversion, the value of magnetic field should be tuned with precision of few nT to a resonance value, so that Ω B = V n with the precision of 10 −16 eV or so. Let us noticed, that since oscillation probabilities of + and − polarization states are different, see eq. (12), resonance can occur for only for one polarization.
Levels of potential energy V n corresponding to quasi-free conditions for n →n observation time ∆t = 0.1 and 1.0 s are shown in the Fig. 2 together with Ω B corresponding to the magnetic field 1 nT. We see that V n can exceed the limit of quasi-free condition in the range of λ between ∼ 10 4 − 10 13 m. In this region n →n oscillation can be suppressed.
However, tiny fifth forces have no effect in intranuclear n →n transformations. One can envisage scenario where n →n will be discovered in intranuclear transformations in large underground experiments although it will not be observed in transformation with free neutrons at the corresponding level, e.g. at the ESS. This can be an indication that some extra potential different between the neutron and antineutron is in play, which can be induced by B − L photons under considerations. This situation can be checked by applying in free neutron experiments the magnetic field with programmed magnitude and direction in the whole neutron flight path and by varying of this field to find the resonance value for which it would compensate the effect of B − L field induced potential V n . Example of such variation of magnetic field is shown in Fig. 3 assuming that V n = 10 −12 eV.

Low scale seesaw model
Is it possible to built a consistent model in which baryon number, or B − L, spontaneously breaks at rather low scales in which case the baryophoton couplings to the neutron can have an effect on the laboratory search of n −n oscillation?
One can discuss a simple seesaw-like scenario for generation of terms (3), along the lines suggested in ref. [4,12]. Let us introduce gauge singlet Weyl fermions, N with Q = −1 and N with Q = 1. These two together form a heavy Dirac particle with a large mass M D . Both N and N can be coupled to scalar χ (Q = 2) and get the Majorana mass terms ∼ χ = υ χ , from the VEV of the latter. We introduce also a color-triplet scalar S, with mass M S with Q = −2/3, having precisely the same gauge quantum numbers as the right down-quark d (R) . Consider now the Lagrangian terms In this way, diagram shown in Fig. 4 Low scale baryon number violation was suggested in Ref. [4], in a model which was mainly designed for inducing neutron -mirror neutron oscillation n − n . This model treats N and N states symmetrically: their Majorana massesM andM are equal, while in addition to couplings (14), there are terms that couple N to u , d and S states from hidden mirror sector with a particle content identical to that of ordinary one (for review, see e.g. [25]). Hence, the lower diagramm of Fig which corresponds to n − n oscillation time τ nn ∼ 1 s. Hence, in this case n − n mixing should be a dominant effect, since two sector share the common Q = B − L. between ordinary and mirror particles, while n −n mixing which breaks Q is suppressed by the small VEV υ χ : Therefore, assuming that ε nn < 10 −15 eV and M D > 1 TeV, for obtaining ε nn > 10 −25 eV one needs υ χ > 100 eV. In this case the Galactic contribution in V n becomes irrelevant, but the possibility of having λ < 1 AU remains robust. Let us remark, that since n − n mixing conserves Q = B − L, baryophotons interact symmetrically with ordinary and mirror neutrons, and thus should have no effect on n − n oscillation. As a matter of fact, n−n mixing can indeed be much larger than n−n. Existing experimental limits on n−n transition allow the neutron−mirror neutron oscillation time to be less than the neutron lifetime, with interesting implications for astrophysics and particle phenomenology [4,27].

Conclusions
Neutron -antineutron transformation searched with free neutrons can be suppressed by the presence of the vector field of baryophotons coupled to B − L charges. Due to assumed baryon number non-conservation these photons should be massive with the mass in the range 10 −11 − 10 −21 eV. This corresponds to a possible region of B − L potential that is not excluded by experimental tests of weak equivalence principle (WEP) so that it could suppress the free neutron n →n transformations. However, if one learns from nuclear instability search that n →n transformation exists but it is suppressed for free neutrons, then this suppression in principle can be removed by the tuning of external magnetic field in the experiment. Weaker B −L fields inducing the potential energy smaller than 10 −16 eV, i.e. below the quasi-free condition limit, practically will not be sensed by n →n transformation and therefore cannot be observed in this way. STEP experiment for Satellite Test of the Equivalence Principle [28] proposed some years ago claimed the sensitivity of WEP testing to the level 10 −18 . STEP mission was not pursued. Corresponding level of magnitude of B − L potential energy that could be excluded in STEP test are also shown in Figure 2.
Let us remark about the possibility of the kinetic mixing of B − L photons with the regular QED photons. Such mixing could make the Equivalence Principle tests potentially different for electrically neutral and charged objects, e.g. neutrons and also neutrinos having non-zero B−L could acquire also the tiny electric charges. As a matter of fact, the considered B − L potentials can have no effect on the oscillations between three neutrinos ν e , ν µ and ν τ since their B − L charges are equal, but they can be relevant for the active-sterile neutrino (e.g. mirror neutrino) oscillations and can suppress them in certain situations.
Also, B − L charge of the Earth would create a B − L magnetic field due to the Earth rotation. Question is whether this can lead to any observable effect?
Concluding, if the neutron-antineutron oscillation will be discovered in free neutron oscillation experiments, this will imply limits on B − L photon coupling constant and interaction radius which are considerably stronger than present limits form the tests of the equivalence principle. The potential V induced by these forces can be excluded down to the values of about 10 −16 eV, independently on the interaction radius λ of these baryophotons. Instead, if n −n oscillation will be discovered via nuclear instability, but not in free neutron oscillations in corresponding level, this would indicate towatds the presence of fifth-force mediated by such baryophotons.

Acknowledgments
Z.B. and Y.K. thank Arkady Vainshtein for useful discussions. The work of A.A. and Z.B. was partially supported by the MIUR triennal grant for the Research Projects of National Interest PRIN 2012CPPYP7 "Astroparticle Physics", and the work of Y.K. was supported in part by US DOE Grant de-sc0014558. This work was reported by Y.K. at the 3rd Workshop "NNbar at ESS", 27-28 August 2015, Gothenburg, Sweden.
Note Added: After this work was completed, it was communicated to us by R. N. Mohapatra and K. S. Babu that they are preparing the work on similar subject.