# Neutron Stars with Baryon Number Violation, Probing Dark Sectors

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## Abstract

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## 1. Introduction

## 2. Setting the Stage—The Neutron Lifetime Anomaly

#### Constraints from Empirical Studies of Neutron $\beta $ Decay within the SM

## 3. Neutron Stars with Baryon Number Violation

#### 3.1. General Conditions

- The new final state particles participate in annihilation or decay channels to yield particles already present plus neutrinos and photons.
- Their production rate (${\mathsf{\Gamma}}_{\mathrm{BNV}}$) is much less than their elimination rate via annihilation ${\mathsf{\Gamma}}_{\mathrm{ann}}$ or decay ${\mathsf{\Gamma}}_{\mathrm{dec}}$.

#### 3.2. Effects of Slow BNV Perturbations

#### 3.3. Constraining BNV from Neutron Star Observations

**PSR B1913+16:**This binary system (Hulse–Taylor binary) is the first binary pulsar ever discovered [139], and consists of a neutron star (${M}_{c}=1.39\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$) and a pulsar (${M}_{p}=1.44\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$) with a pulse period of 59 ms. We use the results from the analysis in Ref. [140] which is based on timing measurements performed over the last 35 years.**PSR J0737−3039A/B:**The only known double pulsar was discovered in 2003 [141], and is comprised of two radio pulsars (A and B) with masses ${M}_{A}=1.34\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$, ${M}_{B}=1.25\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$, and with pulse periods of $22.7$ ms and $2.8$ ms, respectively. We use Ref. [138] which is based on data acquired over 16 years of observation. As a result of the increased accuracy in measurements, the higher-order GR corrections (3.5PN) to ${\dot{P}}_{b}^{\mathrm{GR}}$, and contribution to ${\dot{P}}_{b}^{\dot{E}}$ from the spin-down of pulsar A are added [138].**PSR J1713+0747:**This binary system was discovered in 1993 [142], and it contains a $4.6$ ms radio pulsar with $M=1.3\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ and a companion white dwarf with $M=0.29\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ [143]. A sub-microsecond precision is achieved at measuring its pulse time of arrivals [143] owing to the short spin period and its narrow profile. It has a much longer orbital period (${P}_{b}=67.8$ day) compared to the other two binaries considered here. This is why the inferred limit on ${({\dot{P}}_{b}/{P}_{b})}^{\dot{E}}$ is one order of magnitude better than the limit from the Hulse–Taylor binary despite the higher precision in the latter.

## 4. Implications of Apparent BNV

## 5. Implications of Explicit BNV

#### 5.1. Processes with $|\Delta B|=1$

#### 5.2. Processes with $|\Delta B|=2$

- Explicit and apparent BNV are not mutually exclusive. If mirror neutrons exist and $|\Delta B|=2$ dynamics are operative, then one can consider simultaneous oscillations among all four of $n,\phantom{\rule{0.166667em}{0ex}}\overline{n},\phantom{\rule{0.166667em}{0ex}}{n}^{\prime},\phantom{\rule{0.166667em}{0ex}}{\overline{n}}^{\prime}$. It has been proposed that $n\overline{n}$ oscillations may operate through a “shortcut” [273] through the mirror neutron, via $n\to {n}^{\prime}/{\overline{n}}^{\prime}\to \overline{n}$: these transitions may occur even if the $n\overline{n}$ matrix element is small. This framework can be difficult to test due to the ill-constrained Hamiltonian involving the interactions of the mirror neutron. Proposals exist to probe this scenario at the European Spallation Source [265], at Oak Ridge National Laboratory [274], and at the Paul Scherrer Institute [275].
- Equation (55) assumes that the n and $\overline{n}$ are exactly degenerate in the experimental environment. There are several reasons why this would not be the case; among these are the presence of external matter and magnetic fields [276,277], which lift the degeneracy and suppress oscillations. While these suppress oscillations, they may also stimulate $|\Delta B|=2$ phenomena via other mechanisms, such as neutron-antineutron conversion via scattering, such as $n{e}^{-}\to \overline{n}{e}^{-}$. This may be generated either by long-distance contributions (as in, e.g., Ref. [278]) or through short-distance contributions in SMEFT—these processes can select different subsets of operators from those that generate $n\overline{n}$ oscillations directly. The latter come with higher inverse powers of the new scale; this may allow for ${\mathsf{\Lambda}}_{|\Delta B|=2}$ to be lowered to the $\mathcal{O}(1-10)$ TeV scale without running aground of existing constraints.

#### 5.3. Connections to Lepton Number Violation

#### 5.4. Effects in Neutron Stars

- Processes that destroy two nucleons. These include processes such as $nn\to 2\gamma ,\phantom{\rule{0.166667em}{0ex}}3\nu $, etc. This is completely analogous to dinucleon decay, discussed above.
- Processes that convert nucleons to antinucleons. These include scattering processes such as ${e}^{-}n\to {e}^{-}\overline{n}$.

- Those produced directly in the BNV reaction.
- Those produced by the weak reactions that restore chemical equilibrium after some BNV process has disrupted it (i.e., Urca reactions).
- Those emitted as a result of the heating of the star, via processes such as $NN\to NN\nu \overline{\nu}$.

## 6. Implications of Spontaneous BNV

#### 6.1. Laboratory Constraints on a New Gauge Boson

- This new interaction would also change energy levels, relative to QED predictions, of the antiproton-helium ($\overline{p}$-He) bound state. The constraint from Ref. [338] is shown in blue.
- New forces also change the charge radii and binding energies of nuclei. Ref. [339] studies the effects of new nuclear-range interactions on ${}^{48}$Ca, ${}^{120}$Sn, and ${}^{208}$Pb; their constraint is shown in light blue.
- This interaction would modify the long-distance potential between two protons in the Sun, thereby altering the rate of solar fusion. This could then (1) change the inferred age of the Sun to be inconsistent with the age of the solar system, and (2) modify solar neutrino production to be inconsistent with observations. Constraints have been derived in Ref. [340]. These are shown in shades of pink, corresponding to different proton energies: 10 (light), 50 (medium), and 100 (dark) keV.

- The meson decays radiatively to $\gamma $ and X, e.g., $\eta \to \gamma X$. The computed rate of this process is scheme dependent: rates calculated at the quark level are different than rates calculated in, e.g., the vector meson dominance (VMD) scheme; see Appendix A.1 of Ref. [330].
- The X then decays into some observable final state. At tree level, X can decay to, e.g., ${\pi}^{0}\gamma $ or $3\pi $. If there exists some nontrivial kinetic mixing with $\gamma $, then X may also decay into dilepton pairs, ${e}^{+}{e}^{-}$ or ${\mu}^{+}{\mu}^{-}$, even though these are uncharged under B. Additionally, because X only couples to isoscalar currents, tree-level decays to ${\pi}^{+}{\pi}^{-}$ are absent, barring either (1) nonzero kinetic mixing, or (2) more complicated gauge structures.

#### 6.2. Effects in Neutron Stars—Heavy X

- Neutron stars can be more massive. The increase in the energy density of a given fluid element, relative to some nominal prediction, partially offsets some of the gravitational binding energy of the neutron star, resulting in a heavier star. The heaviest confirmed neutron star, PSR J0740+6620, was initially determined to have a mass $2.{14}_{-0.09}^{+0.10}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ [397], though this has since been refined to $2.08\pm 0.07\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ [398]; any candidate EoS must be stiff enough to support neutron stars at least this heavy.
- Neutron stars have larger radii. As the EoS is stiffened, the increase in pressure makes nuclear matter harder to compress. Therefore, a neutron star of a fixed gravitational mass will be physically larger with a stiffer EoS.

#### 6.3. Effects in Neutron Stars—(Ultra-)Light X

- a flux of the new state, which one could hope to observe directly; or

## 7. Other Imprints of DM Physics on NSs and Their Mergers

## 8. Summary

- In the presence of BNV, the distribution of neutron star masses to be found through gravitational wave studies can be expected to change with lookback time. We note that over the local volume available to us with present and next-generation gravitational wave detectors, the population of stars available to form neutron stars should differ little, making shifts in the mass distribution of the ensemble sensitive to the possibility of BNV effects, albeit likely apparent ones.
- It is possible that BNV can produce unbearably light neutron stars, leading to explosions with detectable signatures in X-rays or soft gamma rays [120]. This may be difficult to realize, however, as common mechanisms of neutron star formation favor roughly $\mathcal{O}\left(1\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}\right)$ stars and as BNV may become inefficient, due to its possible density dependence, in the lightest mass neutron stars.
- We can hope to detect neutron stars of sub-solar mass. This may speak to new mechanisms for neutron star formation and possibly, too, to BNV.
- The possibility of compact objects that are bright in X-ray or neutrino emission may allow the detection of these objects individually, or more probably, through their additional contributions to the diffuse supernova neutrino background, which may soon be detected at Super-K [492]. This effect has also been suggested from considerations of binary-star interactions [493].
- BNV processes with final states involving photons, mesons, and charged leptons (e.g., $n\to {e}^{\mp}{\pi}^{\pm}$) can be expected to dump all of their energy back into the NS, raising the temperature of the NS to a potentially detectable level, given upcoming observational possibilities, both in X-ray and the optical [494]. Ground-based follow-up optical studies of targets of opportunity from gravitational wave observations, given their expected sensitivity [495], may also yield new surprises. Put more pithily, old neutron stars should be cold; if they are not, then this would really be quite a coup.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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