PHYSICAL REVIEW D 101, 083020 (2020)
DAMA/LIBRA annual modulation and axion quark nugget dark matter model
Ariel Zhitnitsky Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
(Received 22 September 2019; accepted 6 April 2020; published 16 April 2020)
The DAMA/LIBRA (DL) experiment shows 9.5σ evidence for an annual modulation in the (1–6) keV energy range, strongly suggesting that the observed modulation has a dark matter origin. However, the conventional interpretation in terms of weakly interacting massive particle–nucleon interaction is excluded by other experiments. We propose an alternative source of modulation in the form of neutrons, which have been liberated from surrounding material. Our computations are based on the so-called axion quark nugget (AQN) dark matter model, which was originally invented long ago to explain the similarity between the Ω ∼ Ω dark and visible cosmological matter densities, i.e., dark visible. In our proposal, the annual modulation is shown to be generated in the keVenergy range, which is consistent with the DL observation in (1–6) keV range. This keV energy scale in our proposal is mostly determined by spectral properties of the neutrinos emitted by the AQN dark matter particles, while the absence of the modulation with energies above 6 keV is explained by a sharp cutoff in the neutrino’s energy spectrum at ∼15 MeV. This proposal can be directly tested by COSINE-100, ANAIS-112, CYGNO, and other similar experiments. It can be also tested by studying the correlations between the signals from these experiments and the signatures from drastically different detectors designed for studies of infrasonic or seismic events using such instruments as distributed acoustic sensing.
DOI: 10.1103/PhysRevD.101.083020
I. INTRODUCTION have been liberated from material surrounding the detector) may be responsible for the observed annual modulation. The DAMA/LIBRA (DL) experiment [1–4] claims the Our proposal is drastically different from previous observation for an annual modulation in the (1–6) keV suggestions [7–10] in one crucial aspect: The neutrons energy range at 9.5σ C.L. The C.L. even higher (12.9σ) for in our case are induced by neutrinos emitted by the axion the (2–6) keV energy range when DAMA/NaI and DL- quark nugget (AQN) dark matter particles. Therefore, the phase 1 can be combined with DL-phase 2 results. The annual modulation observed by DL has a truly genuine DM measured period (0.999 0.001 yr) phase (145 5) origin, though it is manifested indirectly in our framework strongly indicate a dark matter (DM) origin of the modu- through the following chain: lation, the phenomenon which was originally suggested in Ref. [5]; see also Ref. [6]. However, the annual modulation observed by DL is AQN → ðneutrinosÞ → ðsurrounding neutronsÞ → DL: excluded by other direct detection experiments if inter- ð1Þ preted in terms of the weakly interacting massive particle– nuclei interactions. This motivated a number of alternative In this framework, the modulation should obviously show a explanations for the DL signal. In the present work, we proper period of 1 yr with a proper phase as the neutrinos argue that the modulation observed by DL is due to the from Eq. (1) are originated from dark matter nuggets, and neutrons surrounding the detector. In this respect, our the corresponding time variation will be obviously trans- proposal is similar to the previous suggestions [7–10], ferred to the modification of the neutron flux, which where the authors argued that the induced neutrons (which eventually generates the modulation observed by DL. One should emphasize that the emission features of the neutrinos emitted by AQNs such as the intensity and Published by the American Physical Society under the terms of spectrum [which ultimately determines the (1–6) keV the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to energy recoil for the observed DL annual modulation] the author(s) and the published article’s title, journal citation, have been computed in the AQN model long ago for and DOI. Funded by SCOAP3. completely different purposes, and we will use exactly the
2470-0010=2020=101(8)=083020(20) 083020-1 Published by the American Physical Society ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) same original parameters of the model without any inten- hBi, which enters all the observables. Finally, in Sec. II C, tion to modify them to fit the DL observations. we highlight a number of other model-dependent properties, We overview the basic ideas of the AQN model in Sec. II. such as ionization features of the AQNs, their survival One should emphasize that this model is consistent with all pattern, their annihilation rate, and some other questions available cosmological, astrophysical, satellite, and ground- which are relevant for the present studies. based constraints, where AQNs could leave a detectable electromagnetic signature. While the model was initially Ω ∼ Ω A. Generic features of the AQN model invented to explain the observed relation dark visible,it may also explain a number of other (naively unrelated) The idea that dark matter may take the form of composite phenomena, such as the excess of galactic emission in objects of Standard Model quarks in a novel phase goes different frequency bands. The AQN model may also resolve back to quark nuggets [21], strangelets [22], and nuclear- other, naively unrelated astrophysical mysteries, which ities [23]; see also the review in Ref. [24] with a large include, but are not limited to, the so-called “primordial number of references on the original results. In the models lithium puzzle” [11], the so-called “Solar corona mystery” [21–24], the presence of a strange quark stabilizes the quark [12,13], the recent Experiment to Detect the Global Epoch of matter at sufficiently high densities, allowing strangelets Reionization Signature (EDGES) observations [14], and being formed in the early Universe to remain stable over unexpected annual modulation in x rays in the (2–6) keV cosmological timescales. energy band observed by the XMM-Newton observatory The AQN model advocated in Ref. [25] is conceptually [15], among many others. These cosmological puzzles could similar, with the nuggets being composed of a high-density be resolved within the AQN framework with the same set of color superconducting (CS) phase. As with other high-mass physical parameters to be used in the present work to explain dark matter candidates [21–24], these objects are “cosmo- the annual modulation observed by DL, without fitting or logically dark” not through the weakness of their inter- modifications of any of them. actions but due to their small cross section to mass ratio. As Our main results are formulated in Sec. III, where we a result, the corresponding constraints on this type of dark estimate the intensity of modulation due to the neutrino flux matter place a lower bound on their mass rather than the emitted by dark matter nuggets when they traverse through coupling constant. Earth. This section is separated to four different subsec- There are several additional elements in the AQN model tions, Secs. III A, III B, III C, and III D, according to four in comparison with the older well-known and well-studied different elements of the proposal (1). We use precisely the constructions [21–24]. same set of parameters obtained previously for a different (i) First, there is an additional stabilization factor pro- purpose, as reviewed in the Appendix A. We further vided by the axion domain walls (with a QCD estimate the neutrino-induced neutron’s flux in surrounding substructure) which are copiously produced during rocks, which according to the proposal (1) is the source of the QCD transition and which help to alleviate a the observed DL modulation signal. number of the problems inherent in the older models.1 In Sec. IV, we comment on DL arguments [1–4] sug- (ii) The core of the AQN is in the CS phase, which gesting the irrelevance of the induced neutron flux (due to implies that the two systems (CS and hadronic) can muons and neutrinos). We explicitly show why DL argu- coexist only in the presence of the external pressure ments are not applicable for our proposal (1) with its truly which is provided by the axion domain wall. It genuine DM nature, though manifested indirectly. should be contrasted with original models [21–24] In Sec. VA, we comment on a number of experiments which must be stable at zero external pressure. which exclude the DL results, while in Sec. VB,wemakea (iii) Another crucial additional element in the proposal is few comments on recent results by COSINE-100 [16–18], that the nuggets could be made of matter as well as ANAIS-112 [19], and the recent proposal CYGNO [20], antimatter in this framework. which have been largely motivated by the DL observations. The direct consequence of this feature is that the dark Ω Ω Finally, in Secs. VCand VD, we suggest a few novel tests matter density dark and the baryonic matter density visible which could unambiguously support or rule out the pro- posal (1). 1In particular, in the original proposal the first-order phase transition was the required feature of the construction. However, II. AXION QUARK NUGGET DM MODEL it is known that the QCD transition is a crossover rather than the first-order phase transition. It should be contrasted with the AQN In this section, we overview the AQN model. Specifically, framework when the first-order phase transition is not required, as in Sec. II A, we list the most important and very generic the axion domain wall plays the role of the squeezer. Further- – features of the framework, which do not depend on specific more, it had been argued that the nuggets [21 24] are likely to evaporate on the Hubble timescale even if they were formed. In parameters of the model. In Sec. II B, we overview the the AQN framework, the fast evaporation arguments do not apply, observational constraints on a single fundamental parameter because the vacuum ground state energies in the CS and hadronic of this framework, the average baryon charge of the nugget phases are drastically different.
083020-2 DAMA/LIBRA ANNUAL MODULATION AND AXION QUARK … PHYS. REV. D 101, 083020 (2020) will automatically assume the same order of magnitude represents a precise mechanism of how the “baryogenesis” Ω ∼ Ω dark visible without any fine-tunings and irrespectively can be replaced by the charge separation processes in the to any specific details of the model, such as the axion mass AQN framework. ma or size of the nuggets R. Precisely this fundamental The remaining antibaryons in the early Universe plasma consequence of the model was the main motivation for its then annihilate away, leaving only the baryons whose anti- construction. matter counterparts are bound in the excess of antiquark The presence of a large amount of antimatter in the form of nuggets and are, thus, unavailable for fast annihilation. All high-density AQNs leads to many observable consequences asymmetry effects are of the order of one, which eventually as a result of possible (but, in general, very rare) annihilation results in similarities for all components, visible and dark, i.e., events between antiquarks from AQNs and baryons from the Ω ∼ Ω ; Ω ≈ ½Ω þ Ω ; ð Þ visible Universe. We highlight below the basic results, dark visible dark N N¯ 2 accomplishments, and constraints of this model. Λ Let us recapitulate the original motivation for this model: as they are both proportional to the same fundamental QCD It is commonly assumed that the Universe began in a scale, and they both are originated at the same QCD epoch. In Ω ≃ symmetric state with zero global baryonic charge and later particular, the observed matter to dark matter ratio dark 5 Ω (through some baryon-number-violating process, nonequi- · visible corresponds to a scenario in which the number of librium dynamics, and CP violation effects, realizing three antinuggets is larger than the number of nuggets by a factor of famous Sakharov’s criteria) evolved into a state with a net ðΩN¯ =ΩNÞ ∼ 3=2 at the end of nugget formation. It is positive baryon number. important to emphasize that the AQN mechanism is not As an alternative to this scenario, we advocate a model in sensitive to the axion mass ma and it is capable to saturate (2) which “baryogenesis” is actually a charge separation itself without any other additional contributors. It should be (rather than charge generation) process in which the global contrasted with conventional axion production mechanisms baryon number of the Universe remains zero at all times. In Ω ∼m−7=6 when the corresponding contribution scales as axion a . this model, the unobserved antibaryons come to comprise This scaling unambiguously implies that the axion mass must the dark matter in the form of dense nuggets of antiquarks −5 be fine-tuned ma ≃ 10 eV to saturate the DM density today, and gluons in the CS phase. The result of this “charge Ω while a larger axion mass will contribute very little to dark. separation” process is two populations of AQNs carrying a Ω The relative role between the direct axion contribution axion positive and a negative baryon number. In other words, the Ω m and the AQN contribution to dark as a function of mass a AQN may be formed of either matter or antimatter. has been evaluated in Ref. [45]; see Fig. 5 in that paper. CP However, due to the global violating processes asso- Unlike conventional dark matter candidates, such as ciated with θ0 ≠ 0 during the early formation stage, the 2 weakly interacting massive particles (WIMPs), the dark number of nuggets and antinuggets will be different. This matter or antimatter nuggets are strongly interacting and difference is always an order of one effect, irrespectively to macroscopically large. However, they do not contradict any m the parameters of the theory, the axion mass a, or the of the many known observational constraints on dark θ initial misalignment angle 0. We refer to the original matter or antimatter due to the following main reasons [47]. – papers [43 46] devoted to the specific questions related to (i) They are absolutely stable configurations on the ’ the nugget s formation, generation of the baryon asymme- cosmological scale, as the pressure due to the axion try, and survival pattern of the nuggets during the evolution domain wall (with the QCD substructure) is equili- in the early Universe with its unfriendly environment. The brated by the Fermi pressure. Furthermore, it has only comment we would like to make here is that the been shown that the AQNs survive the unfriendly Ω Ω disparity between nuggets N and antinuggets N¯ gen- environment of the early Universe, before and after CP erated due to violation unambiguously implies that the the big bang nucleosynthesis (BBN) epoch [46]. The Ω baryon contribution B must be the same order of majority of the AQNs also survive such violent Ω Ω magnitude as N¯ and N, because all these contributions events as galaxy formation and star formation. are proportional to one and the same fundamental dimen- (ii) They carry a huge (anti)baryon charge jBj ≳ 1025 Λ sional parameter QCD. If these processes are not funda- and so have an extremely tiny number density. Ω Ω mentally related, the two components dark and visible (iii) The nuggets have nuclear densities, so their effective could easily assume drastically different scales. This cross section σ ∼ R2 is small relative to its mass, σ=M ∼ 10−10 cm2=g. This key ratio is well below 2This source of strong CP violation is no longer available at the the typical astrophysical and cosmological limits present epoch as a result of the dynamics of the axion, which which are on the order of σ=M < 1 cm2=g. CP remains the most compelling resolution of the strong problem; (iv) They have a large binding energy such that the see original papers on Peccei-Quinn symmetry [26], Weinberg- Wilczek axion [27,28], Kim-Shifman-Vainshtein-Zakharov invis- baryon charge locked in the nuggets is not available ible axion [29,30], and Dine-Fischler-Srednicki-Zhitnitsky invis- to participate in BBN at T ∼ 0.1 MeV, and the basic ible axion [31,32] models. See also recent reviews [33–42]. BBN picture holds, with possible small corrections
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of the order of ∼10−10 which may, in fact, resolve millisecond range in order to study the correlated signals the primordial lithium puzzle [11]. from AQNs. The modern cosmic ray detectors are not (v) The nuggets completely decouple from photons as a designed to analyze such long time correlations, and result of the small σ=M ∼ B−1=3 ≪ 1 ratio, such that normally such signals would be rejected as a background. the conventional picture for structure formation The strongest direct detection limit is set by the IceCube holds. Observatory’s nondetection of a nonrelativistic magnetic (vi) The nuggets do not modify conventional cosmic monopole [49]. While the magnetic monopoles and the microwave background analysis, with possible small AQNs interact with material of the detector in very different corrections which, in fact, may resolve a tension [14] ways, in both cases the interaction leads to electromagnetic between the standard prediction and EDGES ob- and hadronic cascades along the trajectory of the AQN (or servation (stronger than anticipated 21 cm absorp- magnetic monopole) which must be observed by the tion features). detector if such an event occurs. The nonobservation of To reiterate, the weakness of the visible-dark matter inter- any such cascades puts the following limit on the flux of action is achieved in this model due to the small geometrical heavy nonrelativistic particles passing through the detector: σ=M ∼ B−1=3 factor rather than due to a weak coupling of a hBi > 3 × 1024 ½direct ðnonÞdetection constraint ; ð3Þ new fundamental field to Standard Model particles. In other words, this small effective interaction ∼σ=M ∼ B−1=3 repla- where we assume 100% efficiency of the observation ces a conventional requirement of sufficiently weak inter- of the AQNs passing through IceCube Observatory; see actions of the visible matter with WIMPs. Appendix A in Ref. [50]. Similar limits are also derivable from the Antarctic Impulsive Transient Antenna (ANITA) [51], though this B. Mass distribution constraints result depends on the details of radio band emissivity of the One should emphasize that the AQN construction does not AQN. In the same work, the author also derives the constraint specify the size of the nuggets. In general, for a given axion arising from a potential contribution of the AQN annihilation mass ma there is always a range of B when a stable solution events to Earth’s energy budget which requires jBj > 2.6 × exists [25]. The average size of the nugget within this stability 1024 −1 [51], which is also consistent with Eq. (3). There is also a range scales as R ∝ ma , while the baryon charge of the AQN constraint on the flux of heavy dark matter with mass M< 3 −3 itself scales as hBi ∝ R ∝ ma . In the AQN framework we 55 g based on the nondetection of etching tracks in ancient treat hBi as a fundamental parameter to be constrained mica [52]. It slightly touches the lower bound of the allowed observationally. It is clear that larger hBi values produce window (3) but does not strongly constraint the entire weaker observational signals as σ=M ∼ B−1=3. Furthermore, window (5), because the dominant portion of the AQNs lies any such consequences assume the largest values where the well above its lower limit (3) assuming the mass distribution densities of both visible and dark matter are sufficiently high (6) as discussed below. such as in the core of the Galaxy, the early Universe, or the The authors of Ref. [53] use the Apollo data to constraint stars and planets. In other words, the nuggets behave like the abundance of quark nuggets in the region of 10 kg to conventional cold dark matter in environments where the one ton. It has been argued that the contribution of such density of the visible matter is small, while they become very heavy nuggets must be at least an order of magnitude interacting and radiating objects (i.e., effectively become less than would saturate the dark matter in the Solar visible matter) if they enter an environment of sufficiently neighborhood [53]. Assuming that the AQNs saturate the large density. dark matter, the constraint [53] can be reinterpreted that at As mentioned above, the flux of AQNs on Earth’s surface least 90% of the AQNs must have masses below 10 kg. This is scaled by a factor of B−1 and is thus suppressed for large constraint can be approximately expressed in terms of the nuggets; see Eq. (8) below. For this reason, the experiments baryon charge as follows: most relevant to AQN detection are not the conventional hBi ≲ 1028 ½Apollo constraint : ð4Þ high-sensitivity dark matter searches but detectors with the largest possible search area. For example, it has been Therefore, indirect observational constraints (3) and (4) proposed [48] that large-scale cosmic ray detectors such suggest that, if the AQNs exist and saturate the dark matter as the Auger observatory of Telescope Array may be density today, the dominant portion of them must reside in sensitive to the flux of AQNs in an interesting mass range. the window: An obvious challenging problem with such studies is that the conventional cosmic ray detectors are designed to analyze 3 × 1024 ≲ hBi ≲ 1028 ½constraints from observations : the time delays measured in microns, as cosmic rays are ð5Þ assumed to be moving with the speed of light, while AQNs v ≃ 10−3c move with nonrelativistic velocity AQN . It obvi- Completely different and independent observations also ously requires the latency time to be measured in the suggest that the galactic spectrum contains several excesses
083020-4 DAMA/LIBRA ANNUAL MODULATION AND AXION QUARK … PHYS. REV. D 101, 083020 (2020) of diffuse emission, the origin of which is not well corresponds (within the AQN framework) to the baryon established and remains to be debated. The best known charge window 3 × 1024 ≲ B ≲ 2 × 1028, which strongly example is the strong galactic 511 keV line. If the nuggets overlaps with all presently available constraints (5) on the have a baryon number in the hBi ∼ 1025 range, they could AQN sizes as reviewed above. Precisely this highly nontrivial offer a potential explanation for several of these diffuse consistency check on size distribution along with the (essen- components. It is a very nontrivial consistency check that the tially model-independent) computation of the total luminosity required hBi to explain these excesses of the galactic diffuse of the EUV radiation ∼1027 erg=s from the Solar corona, emission belongs to the same mass range as reviewed above. which is consistent with observations, gives us a strong For further details, see the original works [54–59] with confidence for the plausibility of the identification (6) explicit computations of the galactic radiation excesses for between the nanoflares conjectured by Parker long ago and various frequencies, including excesses of the diffuse x and γ the AQN annihilation events. rays. In all these cases, photon emission originates from the Encouraged by these consistency checks, we adopted the outer layer of the nuggets known as the electrosphere, and all AQN size distribution (6) with parameter α being extracted intensities in different frequency bands are expressed in from the heating corona studies for all our subsequent terms of a single parameter hBi such that all relative Monte Carlo simulations including the computations of the intensities are unambiguously fixed, because they are AQN flux on Earth as given by (8) to be discussed in great determined by the Standard Model physics. detail in the next section. Yet another AQN-related effect might be intimately “ ” linked to the so-called Solar corona heating mystery. C. A few additional comments The renowned (since 1939) puzzle is that the corona has a temperature T ≃ 106 K, which is 100 times hotter than the Here, we make a few additional comments on basic surface temperature of the Sun, and conventional astro- features of the AQNs which are important for the under- physical sources fail to explain the extreme UV (EUV) and standing of this framework, in general, and for specific soft x-ray radiation from the corona 2000 km above the estimates (1) relevant for DAMA/LIBRA signal, in photosphere. Our comment here is that this puzzle might particular. find its natural resolution within the AQN framework as We start this overview with a mention of the electric recently argued in Refs. [12,13,60]. charge which AQNs may carry while propagating in a In this scenario, the AQNs composed of antiquarks fully medium. It is normally assumed that all types of nuggets – annihilate within the so-called transition region, providing a including the old version models [21 24] are neutral at zero T ¼ 0 total annihilation energy which is very close to the observed temperature , because the electrosphere made of EUV luminosity of 1027 erg=s. The EUV emission is leptons will be always formed even when the quark nuggets assumed to be powered by impulsive heating events known themselves are electrically charged (for example, due to the differences in the quark’s masses). However, the neutrality as nanoflares conjectured by Parker long ago. The physical T ≠ 0 origin of the nanoflares remains unknown. If our identifica- will be lost due to the ionization at , in which case the tion of these nanoflares with the AQN annihilating events is nuggets will esquire the positive charge due to the ionized correct, we may extract the baryon charge distribution electrons, while the antimatter nuggets will esquire the dN=dB negative electric charge due to the ionized positrons. The for the AQNs, because the energy distribution Q dN=dE of the nanoflares has been previously modeled by corresponding charge for AQNs can be estimated as the solar physics people to fit the observations, i.e., follows [11,12]: Z ∞ 4πR2 pffiffiffiffiffiffiffiffiffiffiffiffi dN ∝ E−αdE ∝ B−αdB; ð6Þ Q ≃ 4πR2 nðzÞdz ∼ ðT 2m TÞ; ð Þ 2πα · e 7 z0 where slop parameter α slightly varies for different solar where nðzÞ is the density of the positrons in the electrosphere models.3 It is a highly nontrivial consistency check that the which has been computed in the mean field approximation. typical nanoflare energy range E ≃ ð1022 − 6 × 1025Þ erg nano In this estimate, it is assumed that the positrons with 2 p =ð2meÞ
083020-5 ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) affect the AQN’s trajectory) because Q=M ≪ e=mp. completely annihilated or lost a finite portion of their Nevertheless, a nonvanishing charge Q may play a very baryon charges represent a very tiny portion of all AQNs important role in some circumstances, such as the propa- during the entire evolution of the Universe as estimated gation of AQNs in highly ionized plasma in the Solar corona in Ref. [46]. [12,13,60]. The nonvanishing charge Q may also suppress A final comment we would like to make in this the primordial lithium abundance at T ≃ 20 keV due to the subsection is related to spallation, which represents a very strong attraction between the negatively charged AQNs and common process when heavy nuclei lose their baryon positively charged lithium nuclei [11]. charge as a result of interaction with a medium. In contrast Our next comment is related to the survival pattern of the with conventional nuclei, spallation cannot play any AQNs. Comprehensive studies on this matter can be found essential role in the AQN survival pattern due to several in the original work [46]. The only comment we would like reasons. First of all, the gap Δ in the CS phase is typically in to make here is that the dominant portion of the AQNs will the 100 MeV range, in contrast with conventional nuclear survive the evolution of the Universe. However, a very physics where the binding energy normally is in the few small portion of the AQNs which are gravitationally MeV range. The most important distinct feature, however, captured by stars and planets may drastically decrease is as follows. When a large amount of energy is injected their baryon charges and may even experience complete into a heavy nucleus, spallation takes place and a large annihilation.4 The typical size L of the region of the number of neutrons may be liberated, leading to the medium with density ρ where the complete annihilation decreasing of the baryon charge of the heavy original occurs is estimated as σρL ≃ Bmp, where σ is an effective nucleus. Such a process cannot occur with AQNs, because cross section which could be much larger than naive πR2 all particles which get excited due to the energy injection in due to the long-range Coulomb interaction when Q ≠ 0 the CS phase are the colored objects. Therefore, these according to Eq. (7). This effect plays a very important role elementary excitations cannot enter the hadronic vacuum in the Solar corona [12,13,60]. All nuggets which are where the normal baryons live and must stay inside of the gravitationally captured by the Sun will be completely AQN. Therefore, the AQNs do not suffer from spallation annihilated. processes as heavy nuclei do. This is a direct manifestation One may wonder what happens to the axions from AQNs of the same feature of the AQN construction already which are now liberated and become propagating axions mentioned in footnote 1, which states that the confined with average energy hEi ≃ 1.3ma. A small portion of these hadronic and CS phases have different vacua. The same axions will be converted to photons in the background of feature precludes transformation of the entire star or planet the Solar magnetic field and, in principle, can be observed into a new phase if the AQNs made of matter stop in the on Earth. The effect is very small, though, even if one takes deep interior of the star or planet; see also footnote 4. into account the resonance condition due to the plasma effects in the Solar corona [61]. III. DL MODULATION BY AQNS In the case of the AQN traversing Earth’s interior, only some small portion of the baryon charge will be annihi- This section is separated into four different subsections, lated. The full-scale Monte Carlo simulations suggest that Secs. III A, III B, III C, and III D, according to four on average approximately (10–20)% of the total baryon different elements of the proposal (1). charge will be lost as a result of traversing of the AQNs through Earth’s interior; see the column for ΔB=B in A. AQN flux on Earth Table III in Ref. [50]. The average amount of the lost We start our task with the first element from proposal (1) baryon charge depends, of course, on the size distribution by estimating the AQN hit rate per unit area on Earth’s α ρ and parameter defined by Eq. (6). To avoid confusion, let surface assuming that DM is entirely saturated by the us emphasize one more time that all these AQNs which get nuggets. The relevant rate has been studied previously in Ref. [50] for a different problem of computing the axion 4The nuggets made of matter can be stopped in a dense flux produced by the AQNs. Now we estimate the AQN environment such as neutron stars, on scales of the order of hitting rate assuming conventional dark matter density Lstop, when the number of hits is of the order of B, i.e., ρ ≃ 0.3 GeV cm−3 surrounding Earth. Assuming the 2 stop DM πR L hρi ∼ Bmp. However, in contrast with conventional conventional halo model, one arrives to the following ’ – nugget s models [21 24], the AQNs will not turn an entire star result [50]: into a new phase, because the CS phase in the AQN is supported by external pressure due to the axion domain wall, while the – hN_ i 4 10−2 ρ hv i 1025 original nuggets [21 24] are assumed to be stable objects at zero ≃ × DM AQN : ð Þ external pressure. The phenomenon of collecting the matter 2 2 GeV km 8 4πR⊕ km yr 0.3 3 220 hBi nuggets in the cores of stars or planets might be of interest by cm s itself, but it is not the topic of the present work devoted to antimatter AQNs capable of producing neutrinos as a result of The averaging over all types of AQN trajectories with annihilation processes. different masses MN ≃ mpjBj, with different incident
083020-6 DAMA/LIBRA ANNUAL MODULATION AND AXION QUARK … PHYS. REV. D 101, 083020 (2020) angles, different initial velocities, and different size dis- events. We also need to know the basic features of the tributions does not modify much this estimate. The result neutrino spectrum emitted by AQNs. The lessons from that (8) suggests that the AQNs hit Earth’s surface with a studies can be used for estimations of the neutrino flux from frequency approximately once a day per 1002 km2 area. AQNs, which is the main subject of this subsection. The hitting rate for large size objects is suppressed by the distribution function fðBÞ ∝ B−α as given by Eq. (6). 1. Detour on the axion production The estimate (8) explicitly shows that conventional DM It has been noticed in Ref. [62] that the large number of detectors are too small in size to detect AQNs directly, as 5 axions will be produced, because the axion domain wall the corresponding flux is many orders of magnitude smaller will start to shrink during the AQN annihilation events and than the one due to the conventional WIMPs. However, emit the propagating axions which can be observed. The some modern cosmic ray detectors, such as Pierre Auger corresponding spectrum has been computed in Ref. [61], observatory, in principle, are capable to study small flux of where it has been shown that the emitted axions will have the order of Eq. (8) as suggested in Ref. [48]. One should typical velocities hvai ≃ 0.6c in contrast with conventional also mention that a smaller size IceCube detector imposes a galactic axions characterized by small velocities ∼10−3c direct constraint on the average baryon charge of the nugget hBi ≥ 3 1024 such that these two different production mechanisms can be × ; see Appendix A in Ref. [50]. easily discriminated. From Eq. (8), one can derive the total hit rate for the hN i ’ The average number of the emitted axions a as a entire Earth s surface, which is given by [50] result of the AQN annihilation events in Earth’s interior can ρ hv i 1025 be estimated as follows [63]: hN_ i ≃ 0 67 −1 DM AQN : ð Þ . s GeV km 9 0 3 3 220 hBi hΔm i . cm s AQN hNai ≈ κa ; ð10Þ hEai After the nugget hits the surface, it continues to propagate by annihilating the material along its path. The trajectory of where coefficient κa determines the relative amount of the AQN is a straight line, as only a small portion of the annihilating energy (per unit baryon charge) transferred to momentum (and the baryon charge) will be lost in this the axion production. The computation of the coefficient journey. The energy produced due to the annihilation κa ≃ 1=3 as well as hEai ≃ 1.3ma is a straightforward events will be isotropically dissipating (in the rest frame exercise [61], as it represents a conventional quantum field of the nugget) along the propagation. theory problem for a weakly interacting axion field. The rate (9) includes all types of the AQN’s trajectories The energy flux of the axions (being averaged over all inside Earth’s interior: the trajectories when AQNs hit the emission angles and summed over all trajectories) mea- surface with an incident angle close to 0° (in which case the sured on Earth’s surface is estimated as [63] ’ AQN crosses Earth s core and exits from the opposite site of Earth) as well as the trajectories when AQNs just touch dE v hΔm c2i a ≃ κ a hN_ i AQN ; ð Þ the surface with an incident angle close to 90° (when AQNs a 2 11 dtdA c 2πR⊕ leave the system without many annihilation events deep underground). The result of summation over all these which has a proper dimensionality [GeV · cm−2 ·s−1] for trajectories can be expressed in terms of the average mass the energy flux. The axion flux for this mechanism is hΔm i (energy) loss AQN per AQN. The same information estimated as can also be expressed in terms of the average baryon charge loss per nugget hΔBi, as these two are directly related: dN v hΔm c2i a a _ AQN hΔm i ≈ m hΔBi ≃ κa hNi ; ð12Þ AQN p . The corresponding Monte Carlo sim- dtdA c hE i2πR2 hΔm i a ⊕ ulations with estimates for AQN have been carried out in Ref. [50]; see Table III in that paper. This information which has a proper dimensionality [cm−2 ·s−1] for the will be very important for our analysis in Sec. III B,asit axion flux. provides a normalization for the total neutrino flux being We are now ready to estimate the neutrino emission from ’ emitted by the AQNs when they traverse Earth s interior. the AQNs, which is the main topic of this subsection. One could follow the same logic of computations as highlighted B. Neutrino production from AQNs The second element of the proposal (1) requires the 5As mentioned in Sec. II, the axion domain wall plays the estimation of the neutrino intensity due to the AQN important role of a squeezer stabilizing the nugget. This energy has been accumulated and stored during the QCD epoch at the annihilation processes. Before we proceed with the corre- moment of formation. The corresponding energy accounts for a sponding estimates, we want to make a short detour related considerable portion of the nugget’s total energy, which is to the axion production due to the same AQN annihilation parametrized by κa ≈ 1=3; see below Eq. (10).
083020-7 ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) above, with the only difference is that instead of emission lack of knowledge of the CS phase (in contrast with the of the axions one should study the neutrinos, which are confined phase, where all NG masses and relevant branch- similar to axions, as they can easily propagate through ing ratios are measured with very high accuracy). An Earth’s entire interior. This is because the relevant cross additional uncertainty also comes from the lack of under- section with surrounding material is very small in both standing of the fermion excitations of the CS phases which cases, and formula (11) can be applied for estimation of the may be, in fact, the dominant contributors to the neutrino neutrino flux on Earth’s surface with corresponding fluxes. This is because the NG bosons which are produced _ replacements κa → κν and Ea → Eν, while factor hNi in as a result of annihilation events in the CS phase cannot Eq. (12), of course, remains the same. leave the system and consequently decay to emit neutrinos, as they must stay inside the nuggets.7 It should be 2. On neutrino spectrum in the CS phase contrasted with the hadronic phase when pions and kaons (produced as a result of pp¯ annihilation) decay to muons If we had a conventional hadronic phase inside the and neutrinos in a vacuum. Therefore, the NG bosons in CS nuggets, the corresponding computations of neutrino phases are likely to be absorbed by fermion excitations (if emission would be a very simple exercise. Indeed, we kinematically allowed), which consequently decay to know a typical yield of the pseudo-Nambu-Goldstone (NG) neutrinos. bosons per annihilation event of a single baryon charge Indeed, in unpaired quark matter, neutrino emissivity is pp¯ (such as ). We also know a typical decay pattern for all dominated by the direct Urca processes8 such as NG bosons such as K → μν and π → μν with consequent μ − d → u þ e þ ν¯e. In the case of antinuggets, it should decays. We also know with very high precision the be replaced by antiquarks with the emission of neutrino νe branching ratios for the nonleptonic decays of the NG eþ K → 3π η → 3π and positron with the energy determined by the energy bosons such as and with consequent of the fermion excitation, which itself assumes the energy decays to neutrinos. It would allow us to compute the of the order of the NG mass; see Appendix A3with more total number of neutrinos per single annihilation event. This comments on this matter. would also allow us to compute the energy spectrum of 6 If this process indeed becomes the dominant mechanism neutrinos emitted by the AQNs. of the neutrino emission from AQNs, then one should Unfortunately, we do not have this luxury of knowing all expect that the number of neutrinos per single event of these key features in the CS phase. Therefore, we cannot annihilation should greatly exceed the number of antineu- predict the spectrum of neutrinos emitted by AQNs. The trinos per single event of annihilation, which we assume to only solid and robust information which is available today be the case. Formally, this case can be expressed as follows: is the typical scale of the lightest NG mass in the CS phase, which is normally estimated in the (20–30) MeV range. hΔm c2i hΔm c2i hN i ≈ κ AQN ; hN i ≈ κ AQN ; This scale for NG masses can be translated to an estimation ν ν m c2 ν¯ ν¯ m c2 for a typical neutrino’s energy scale in the AQN framework. p p Assuming that one-half of the lightest NG mass goes to the κν ≫ κν¯ ;Eν ≲ 15 MeV; κν ≳ 1; ð13Þ neutrino’s energy, we expect that hEνi ≲ 15 MeV. While a typical energy scale for the NG mass and the where the coefficients κν and κν¯ describe the number of corresponding neutrino’s energies in the AQN framework is neutrinos and antineutrinos, respectively, produced per established with a reasonable accuracy because it is based single annihilation event, similar to parameter κa ≃ 1=3 on the well-established theory of CS phases [66,67], the entering expression (10) and describing the axion emission computation of the corresponding neutrino spectrum is a due to the AQN annihilation events. much harder problem. The basic problem is, of course, the We conclude this subsection with the following generic comment. Our system is the strongly coupled gauge theory, 6This is precisely the set of assumptions adopted by Ref. [64], the QCD. It should be contrasted with conventional weakly where the authors claimed that dark matter in the form of AQNs coupled gauge theories when all computations are under cannot account for more than 20% of the dark matter density. complete theoretical control. In our system, it is very hard This claim was based on the assumption that the annihilation to predict realistic spectra and intensities for neutrino and events follow the conventional (for a confined phase) pattern, in which case a large number of neutrinos will be produced in the (20–50) MeV range, where the sensitivity of underground 7This is because all excitations in the CS phase are color neutrino detectors such as Super-Kamiokande have their highest objects and cannot propagate in a hadronic vacuum. 8 − signal-to-noise ratio. The basic claim of Ref. [65] is that Another direct Urca process e þ u → d þ νe which for annihilation processes involving an antiquark nugget in the CS antinuggets would correspond to emission of the antineutrino phase proceed in a drastically different way than assumed in is likely to be strongly suppressed, as it requires the presence of Ref. [64] when the lightest pseudo-Nambu-Goldstone meson has positrons with sufficiently high energy above the Fermi surface. a mass in the (20–30) MeV range. As a result of this crucial For a low temperature, the corresponding density for positron difference, the neutrino’s energies will be in the 15 MeV range, excitations is exponentially small ∼ expð−Eeþ =TÞ. This argument well below the present day constraints. suggests that κν ≫ κν¯ as Eq. (13) states.
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“ ” Φ ≃ 8 103 ð −2 −1Þ antineutrino fluxes in the 15 MeV energy range due to a solar hep component is close to νe × cm s variety of possible phases, high sensitivity to the param- [68]. The atmospheric and diffuse supernova neutrino eters, and a large number of possible decay channels backgrounds are at least 2 orders of magnitude smaller producing neutrinos and antineutrons, as we discussed than the hep component and can be safely ignored in our above. Such an analysis could be coined as “the nuclear discussions. physics of CS phases.” The complexity and uncertainties of One should also mention the Super-Kamiokande stringent such studies (though it is entirely based on the Standard Φ < ð1 4–1 9Þ −2 −1 constraint on antineutrino flux ν¯e . . cm s at Model physics) are the main reasons to introduce phenom- E>19 3 Φ < large energies . MeV [69] and constraint ν¯e enological parameters κν and κν¯ in Eq. (13), which will be 4 −2 −1 4 × 10 cm s at smaller energies 8 MeV 083020-9 ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) where we used the AQN-induced neutrino flux (16) and The starting point is the standard formula for energy cross section σ ≃ 10−41 cm2 for the neutrino-induced neu- transfer ΔE as a result of elastic 2 → 2 scattering when a 208 tron spallation for the Pb target. The effective volume V target of mass m2 is at rest, struck by a particle with mass entering Eq. (17) will be discussed later in the text. The m1 with kinetic energy En: AQN-induced neutron rate production per unit volume can m m be represented as follows: ΔE ¼ 2E 1 2 ð1 − θ Þ: ð Þ n ðm þ m Þ2 cos CM 20 1 2 RAQN AQN ν −2 neutron rν ¼ ≃ 10 κ : ð Þ The entire section of Ref. [7] was devoted for an explan- V · ν 3 18 day · m ation of numerous uses and misuses of this formula in different circumstances. We agree with most of the com- This rate is approximately one order of magnitude lower if ments, careful explanation of misconceptions, and detailed κν ≃ 1 in comparison with corresponding estimates adopted analysis of the neutron-nuclear interaction given by Ralston in Refs. [2,10] for the neutron’s rate induced by the solar in Ref. [7]. We defer our specific comments within the neutrinos. It could be the same order of magnitude as used present context until Sec. IV. in Refs. [2,10] if κν ≃ 10; see estimate (13). We return to the Now we want to make a numerical estimate for the recoil significance of estimate (18) later in the text. The only energy using expression (20) and identifying m1 with the comment we would like to make here is that the typical neutron characterized by the kinetic energy (19), while kinetic energy of the neutrons liberated by this mechanism 11 m2 ≃ 23m1 with the lightest Na nucleon from the DL will be in the 102 keV range; see estimate (19) below. detector consisting of 25 radio-pure NaI crystal scintilla- Indeed, the first thing to notice is that the typical neutrino tors: energy (13) is well above the neutron emission threshold E > 7 37 208 ν . MeV for the Pb target such that neutron ΔE ¼ 8.6 keV ð1 − cos θ Þ: ð21Þ spallation is kinematically allowed. Furthermore, if one CM assumes that momentum resulting from the neutron spalla- The significance of this estimate is hard to overstate, as it tion is mostly transferred to the liberated neutron, such that p ≃ ðp0 − p Þ unambiguously shows that the recoil energy cannot exceed n ν ν , the kinetic energy of the neutron can be the value (21), which is amazingly close to the 6 keV cutoff estimated as follows: observed by DL. Furthermore, the scale (19) was not p2 invented for the specific proposal (1), in contrast with E ≃ n ∼ 102 : ð Þ many different WIMP-based suggestions to fit the observed n 2m keV 19 n DL modulations. Rather, this scale is entirely determined by the cutoff in neutrino energy (13), which is itself One should also add that there is a sharp cutoff of the order unambiguously fixed by typical NG masses in the CS ∼102 ’ of keV for the neutron s energy (19) produced by this phase. All these scales have been known for quite some mechanism. It is determined by the neutrino energy (13) in time, as reviewed in Appendix A. the 15 MeV range, which itself is kinematically bound from The key element of our proposal (1) is that the flux of the above by corresponding NG masses in the CS phase as induced neutrons (18) with kinetic energy (19), which reviewed in Appendix A. The presence of such a sharp eventually generates the signal in the DL detector with cutoff will be an important element in our arguments recoil energy (21), is the subject of annual modulation supporting the proposal (1). because all the intensities are proportional to DM velocity What do the neutrons, characterized by the rate (18) and hv i as Eq. (9) states. Therefore, to describe the energy (19), do if they enter the DL detector? This is the AQN corresponding modulations, one can use a conventional subject of the next subsection. formula [5,6]: hv ðtÞi ≃ V þ bV ωðt − t Þ; ð Þ D. DL modulation (1) due to the neutrons AQN ⊙ ⊕ cos 0 22 The fourth element of proposal (1) represents the most −1 controversial portion of our analysis. We shall try to argue where V⊙ ¼ 220 km s is the Sun’s orbital speed around −1 that neutrons characterized by the flux (18) and energy (19) the Galactic center, V⊕ ¼ 29.8 km s is Earth’s orbital −1 may serve as the source of the observed DL annual speed around the Sun, ω ¼ 2π yr is the angular fre- modulation. The corresponding computations are very hard quency of the annual modulation, and jbj ≤ 1 is the v to carry out, as they are inevitably based on nuclear physics geometrical factor associated with the direction of AQN of a large number of very complicated systems. Fortunately, relative to the orbital plane of Earth, t0 ≃ 0.4 yr corre- there are many specific experiments and tests which can be, sponding to June 1. Hence, it is natural to expect that the in principle, performed, to be discussed in Secs. IV and V. modulation must be of the order of OðV⊕=V⊙Þ ∼ 10%,as These tests can support or rule out the proposal (1). incoming flux of the AQN particle explicitly proportional 083020-10 DAMA/LIBRA ANNUAL MODULATION AND AXION QUARK … PHYS. REV. D 101, 083020 (2020) hv ðtÞi to AQN according to Eq. (9). The corresponding value should be considered as a consequence of the complexity of hN_ ðtÞi enters the expression for the AQN-induced neutrino the system. flux (14), eventually generating the neutrons (18). The observed rate (24) matches the AQN-induced The very hard and challenging question remains to be modulation (25) if κν ≃ 10 and L ∼ 10 m. This required answered if this neutron intensity (17) is sufficient for the length L ∼ 10 is definitely much greater than the neutron’s explanation of the observed DL modulation. To answer this mean free path λ ≃ 2.6 m, which was extracted from the question, one has to understand the numerical value for the studies on the muon-induced background [72] and adopted effective volume V entering formula (17). As we already by Refs. [2,10] in the context of the present work on DL mentioned, this is very complicated problem of nuclear modulations. In Sec. IV, we comment on the consistency physics as emphasized and nicely explained in Ref. [7]. (25) with DL observations, while in Sec. V we make a few Therefore, instead of theoretical speculations about the comments on the relation to other experiments. Also in value for the effective volume entering formula (17),we Sec. V, we suggest possible tests (such as the measuring reverse the problem and estimate the volume V which of the spatial directions of moving neutrons along with would match the DL modulation. We suggest several time modulation) which could support or rule out the experiments how this proposal (1) and large rate (17) with proposal (1). effective volume V can be tested in Secs. IV and V. The DL modulation amplitude in terms of counts per day IV. COMMENTS ON DL ARGUMENTS (cpd) [4] reads The DL Collaboration, of course, discussed the possibil- ity that their signal is associated with neutron flux (induced cpd DL modulation rate ¼ð0.0103 0.0008Þ : ð23Þ by muons or neutrinos or both). In fact, the entire paper [2] kg keV was devoted to the analysis of a possible role of neutrino- induced and muon-induced neutrons. These possibilities This rate must be multiplied to 4 keV for the (2–6) keV were discarded in Refs. [1–4] based on the following energy range and 250 kg to get the total modulation rate arguments: (1) modulation phase arguments; counts (2) energy range arguments; and DL total modulation ≃ 10 : ð24Þ day (3) intensity arguments. We want to make a few comments on each of the items On the other hand, assuming that 10% of neutrons (18) are from this list. We start with the modulation arguments, while the energy and intensity arguments will follow. the subject of annual modulation with proper phase t0 as “… explained in Eq. (22), we arrive at the following estimate in (1) Quote from Ref. [2]: It is worth noting that terms of the required effective volume V ≡ L3 which neutrons, muons and solar neutrinos are not com- saturates the DL modulation: peting background when DM annual modulation signature is investigated since in no case they can mimic this signature…” In the proposed scenario neutrons L 3 ≃ κ : ð Þ (1), this argument obviously does not apply, because AQN-induced modulation ν 10 25 day m both the neutrino flux (16) and the neutron flux (17) with typical energy (19) are automatically the sub- A few comments are in order. First of all, parameter κν was ject of the annual modulation (22) with proper phase introduced as a number of neutrinos being produced as a t0 ≃ 0.4 yr corresponding to June 1. This is because result of annihilation of a single baryon charge; see (13).It the source of the modulation in this framework has a could be as small as κν ∼ 1, but it could be as large as truly genuine DM origin represented by AQNs. κν ∼ 10, being consistent with presently available con- (2) DL has carried out comprehensive studies on the straints as mentioned at the end of Sec. III B. The basic dependence of the annual modulation as a function reason for this uncertainty is that our system is the strongly of the energy interval. The claim is that the modu- coupled gauge theory, to be contrasted with conventional lation is not observed above the energy ∼6 keV. In weakly coupled gauge theories when all computations are particular, the modulation amplitude for the energy under complete theoretical control. In our system, it is very above 6 keV for the whole datasets (DAMA/NaI, hard to predict realistic spectra and intensities for neutrino DAMA/LIBRA-phase-1, and DAMA/LIBRA- and antineutrino fluxes in the 15 MeV energy range due to phase-2) is shown to be consistent with zero; see the variety of possible phases, high sensitivity to the Fig. 11 in Ref. [4]. This property is indeed very hard parameters, and large number of possible decay channels to understand in terms of the conventional physics producing neutrinos and antineutrons, as we discussed advocated in Refs. [7–10]. above. This deficiency in our computational power should At the same time, this unique feature of the system not be treated as a weakness of the proposal. Instead, it (characterized by a sharp cutoff at ∼6 keV) 083020-11 ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) automatically emerges in our framework. Indeed, the V. RELATION TO OTHER EXPERIMENTS neutrino flux (16) with typical neutrino energy Eν ∼ AND POSSIBLE FUTURE TESTS 15 MeV is determined in our framework by the NG This section is separated into three different subsections. masses in the CS phase; see (A3). The correspond- First, in Sec. VA, we make a few comments on the previous ing neutrino-induced neutron flux (17) is charac- experiments which exclude the DL signal if interpreted in terized by the typical energy (19) formulated in terms of WIMP-nuclei interactions. We continue with a terms of Eν. The sharp cutoff for the recoil energy more recent analysis in Sec. VB, where we make a few (21) in this framework (which falls into the proper comments on some recent experiments designed to repro- ∼6 keV energy range) is determined by (19), which is essentially determined by the NG masses in the duce (or rule out) the DL signals. Finally, in Secs. VCand CS phase as reviewed in Appendix A. One should VD, we offer a few novel possible tests which can support emphasize that all these energy scales have not been or refute the proposal (1) explaining the DL signal in terms specifically invented in this work to explain the of the AQN-induced neutrons. In particular, in Sec. VDwe observed DL modulations with (1–6) keV energy; propose to study a specific correlation between the ampli- rather, the relevant energy scales have been estab- fication in the AQN-induced neutron flux and the impulses lished long ago in unrelated studies for different of the infrasound and weak seismic waves. purposes in a different context. (3) The neutrino flux (16) originated from AQNs in this A. Previous experiments framework is lower than the background solar We start with a few comments on the experiments which Φ ≃ 5 106 −2 −1 neutrino flux νe × cm s for this energy exclude the DL results. The corresponding collaborations 8 band (originating from solar B) at least by a factor include but are not limited to CDEX [73], CDMS- II of 5 for κν ≃ 10, as mentioned at the end of Sec. III [74,75], EDELWEISS-II [76],LUX[77], SuperCDMS B. The corresponding neutrino-induced neutron flux [78], XENON10 [79], XENON100 [80], and CoGeNT (17) also must be lower in comparison with the [81]. The main claim of these collaborations can be intensity of neutrons induced by the solar neutrinos. formulated as follows: If DL modulation is interpreted in However, the key point is that this subdominant terms of WIMP-nuclei interactions with given σ and given neutron component is originated from AQNs and, m WIMP, then the DL signal is excluded with a very high therefore, is the subject of conventional DM annual level of confidence [82]. t modulation (22) with proper phase 0. From the perspective of the proposal (1), there could be a Is the corresponding neutrino-induced neutron’s inten- 9 number of reasons why DL observes the signal while other sity sufficient to explain DL modulation? DL argued that collaborations do not. First of all, most of the collaborations “ ” – the answer is no [1 4]. However, the DL arguments on (with a few exceptions such as CoGeNT [81] and CDMS- II ’ the neutron s intensity were challenged in Ref. [7].Wehave [75]) did not carry out dedicated studies on the time nothing new to add to these extensive discussions on the modulation, which was the crucial ingredient in DL argu- possible role of neutrino-induced neutrons. Instead of ments. From the AQN perspective, the time modulation is speculations about this very complex nuclear physics the key element when the subdominant neutron flux can system with a complicated resonance structure, we suggest manifest itself if proper time modulation studies are to test the proposal (1) by measuring the modulation, performed. intensity, and directionality in coordinate space of the Another reason (for negative results) could be related to neutrons which are responsible for the recoil (21). different neutron shields used by different collaborations. The subdominant flux of the AQN-induced neutrons can We refer to the paper by Ralston [7], where the subject on be, in principle, discriminated from the dominant compo- the complex behavior of neutrons in a complicated envi- nents, including the solar neutrino-induced neutrons if the ronment is nicely presented. This analysis obviously shows ’ direction of the neutron s momentum and the modulation are that even minor differences in neutron shields may have measured. This is because the solar neutrinos are propagat- dramatic effects and drastically change the impact of the ing from a single direction in the sky, while AQN-induced neutron’s background. neutrinos (which have a truly genuine DM origin) are Yet one more reason, probably the most important one in randomly distributed in space. This topic on possible tests the context of the present work, is as follows. The AQN- of the proposal (1) represents the subject of Sec. V. induced neutrons with energy (19) are scattering off Na in the DL experiment, generating recoil energies which fall – 9 into the (2 6) keV bin according to Eq. (21). The recoil As we already mentioned above, the question is essentially energies for heavier targets such as xenon or germanium in reduced to the quantitative understanding of the effective volume V ¼ L3 which enters (25). Our estimates from the previous different experiments could be below the threshold, section show that, if L ≃ 10 m, the AQN-induced rate (25) because the energies of the time-modulated neutrons matches the observed rate (24) with κν ≃ 10. (19) are bound from above with a cutoff being determined 083020-12 DAMA/LIBRA ANNUAL MODULATION AND AXION QUARK … PHYS. REV. D 101, 083020 (2020) by energies of the AQN-induced neutrinos. The energies of the capability to measure the directionality, which is the these neutrinos are also bound from above and cannot key element of the CYGNO [20] proposal. It is important exceed (13), as they are determined by NG masses in the that it will be located at the same site (LNGS) where DL is CS phase. As we already mentioned previously, all these located. Therefore, the neutron flux must be the same, energy scales in proposal (1) have not been invented to fit including the subdominant AQN-induced component (25) the DL signals. Rather, the relevant energy scales have been which is the subject of annual modulation. The collabo- established long ago in unrelated studies for different ration is planning to measure (initially) the neutron flux and purposes in a different context. its modulation without neutron shielding.11 Such measure- ments may play a key role in supporting (or ruling out) the B. Recent activities proposal (1), because the AQN-induced neutrons are responsible for the recoil (21). The collaboration is also Now we want to make a few comments on recent planning in the future to reach the neutrino floor by dedicated experiments which were specifically designed measuring the neutrinos and their directions. In particular, to test the DL annual modulation signal. It includes CYGNO could discriminate the neutrinos from the Sun by COSINE-100 [16–18], ANAIS-112 [19], and CYGNO identifying their directions. Furthermore, the CYGNO [20] experiments. We also want to mention other types instrument will be capable to determine nuclear recoil of experiments [83–85] which were not originally designed directions, which would allow the collaboration to dis- to test the DL annual modulation signal. However, their criminate WIMP-like DM from AQN-induced events (1). capabilities to measure the directionality could play a Such measurements, if successful, would obviously play an decisive role in detecting the DM signals. To be more important role in supporting (or ruling out) the proposal (1). specific, we choose to mention these experiments due to the In addition to that, the dominant solar neutrinos in the following reasons. energy range Eν ≲ 12 MeV could be discriminated from The COSINE-100 experiment was mostly motivated by subdominant AQN-induced neutrinos (A3). Furthermore, DL annual modulation. The aim of the collaboration is to neutrinos in the energy band Eν ≥ 12 MeV cannot be reproduce (or refute) the signal and to search for a possible originated from the Sun at all as a result of the 8B threshold. origin for the modulation, if observed. The COSINE-100 The discovery of such neutrinos and measuring of their Collaboration uses the same target medium (sodium iodide) annual modulation with an intensity in the range (16) which is the crucial element in the context of the present would be enormous support for the proposal (1), as the flux proposal (1), as recoil energies fall into the (2–6) keV bin in of the atmospheric neutrino is at least 3 orders of magnitude our scenario according to Eq. (21). Presently, the COSINE- lower than (16). 100 data are consistent with both a null hypothesis and a We also want to mention several other experiments DL (2–6) keV best fit value with 68% confidence level. [83–85] which are capable to measure the directionality. More data are obviously needed. It is important that The idea is to use carbon nanotube arrays or graphene COSINE-100 is planning to measure the neutron’s intensity layers to measure the directionality of the DM signals. As and neutron’s modulation [17], in which case COSINE-100 we mentioned above, measurements of the directionality would know if the possible modulation is due to the could play a decisive role in detecting the DM signal. neutrons.10 It is obviously the key element of the proposal (1) based on the subdominant AQN-induced component of neutrons which, however, manifests itself by annual C. Possible future tests modulation. We already mentioned in Sec. VB a few possible tests The ANAIS-112 Collaboration also uses the same target which can support or rule out the proposal (1) with existing medium as DL and COSINE-100. ANAIS-112 has recently or planning experiments: COSINE-100 [16], ANAIS-112 published the first results on annual modulation [19]. Their [19], and CYGNO [20]. In this subsection, we want to best fits are incompatible at 2.5σ with the DL signal. The mention a specific for the AQN framework phenomenon goal is to reach the sensitivity at 3σ level in 5 yr. As the when the intensity of the AQN-induced neutrinos may be ANAIS-112 Collaboration uses the same target material, amplified by a very large factor (up to 104) which greatly our comments from the previous paragraph in the context increases the chance for discovery of such AQN-induced of the present proposal (1) also apply to the ANAIS-112 neutrons which always accompany the neutrinos according experiment, especially as ANAIS-112 and COSINE-100 to Sec. III C. Therefore, this neutrino amplification factor agreed to combine their data. will be obviously accompanied by the corresponding The CYGNO proposal [20] is different from the amplification of the neutrino-induced neutron flux (17) COSINE-100 and the ANAIS-112 experiments due to and (18). 10I thank G. Adhikari for answering a large number of my 11I thank Elisabetta Baracchini for answering a large number of questions during the PATRAS-2019 axion meeting about the my questions during the PATRAS-2019 axion meeting about the future plans of the collaboration. future plans of the collaboration. 083020-13 ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) The idea was originally formulated for the axions in that it becomes the dominant one for a short period of time Ref. [63], and the effect was coined as the “local flash.” The lasting for about 1 s. computations can be easily generalized for the neutrinos in An important lesson to be learned from these estimates is straightforward way; see Appendix B for some technical as follows. A subdominant neutrino flux induced by AQNs details. It can be explained as follows. may become a dominant portion of the neutrinos, over- If the AQN hits the surface at distance d ≪ R⊕ from the passing the solar neutrino flux in this energy band for a very detector, the short-lasting flash occurs with amplification short period of time. Needless to say, this AQN-induced factor AνðdÞ measuring the relative short-lasting spark of neutrino flux (16) and the corresponding neutron flux (17) the neutrino flux with respect to the neutrino flux (16) are also the subject of annual (22) and daily modulations,12 computed by averaging over the entire Earth’s surface over similar to the ones studied for the axion search experiments a long period of time. The amplification AνðdÞ is highly [63]. The measure of directionality and modulation as sensitive to distance d and can be approximated as follows described in Sec. VB may help to discriminate this [see (B4) for the derivation]: subdominant AQN-induced neutrino flux from the domi- 8 nant solar B component. 2 1 R⊕ AνðdÞ ≃ : ð26Þ hN_ ihΔti d D. Search for correlations with other AQN-induced phenomena One should note that the correction to the neutrino flux The flux (8) suggests that the AQNs hit Earth’s surface (B3) due to the traversing of a nearby AQN depends on with a frequency approximately once a day per 1002 km2 κ unknown parameter ν. However, the relative amplification area, which is precisely the source for a short-lasting A ðdÞ ν with respect to the averaged neutrino flux (16) does amplification in the neutrino production discussed in _ not depend on κν as Eq. (26) states. In formula (26), hNi is Sec. VC. It is important to emphasize that such events hΔti ≃ 2R =hv i determined by Eq. (9), while ⊕ AQN is the occur along with other processes which always accompany time for the AQN to cross Earth averaged over the entire the AQN propagation through the atmosphere and Earth’s ensemble of AQN trajectories traversing Earth. underground. Therefore, there will be always some corre- As one can see from Eq. (26), a huge amplification may lations between the amplifications in the neutrino flux and indeed occur for d ≪ R⊕. However, the probability for other associated phenomena in the vicinity of the area such an event to happen is very tiny and can be estimated as where an AQN event occurs. For example, if the DM detector (sensitive to neutrino- 1 1 ≃ ; ð Þ induced neutrons such as DL) and an axion search detector event rate _ 3 1=2 · 3=2 27 d ∼ d ½hNihΔti Aν are localized close to each other on a distance νa , there will be the axion signal due to the amplification Aa and the see Appendix B for details. The “local flash” lasts for a neutrino-induced neutron signal due to the amplification short period of time which can be estimated as follows: Aν. These signals must be correlated in the form of two almost simultaneous short-lasting sparks between these two 2d hΔti 1=2 1 signals in two different detectors. The observation of these τ ≃ ≃ : ð Þ hv i _ 1=2 28 cross-correlated signals (collected during a long period of AQN hNi Aν time and by measuring the directionality in the DM detector We summarize in Table I a few choices of time duration τ to discriminate the background) would unambiguously support the proposal (1) on the nature of the observed and the event rate as a function of amplification factor Aν. In particular, it would be a daily short-lasting “flash” when DL modulation signal. A similar cross-correlation between the intensity of the subdominant AQN-induced neutrino different synchronized axion stations from a global network 2 as suggested in Ref. [86] and a nearby neutrino detector component (16) is amplified by factor AνðdÞ ≃ 10 such would also strongly support the proposal (1). In particular, the position of the Center for Axion TABLE I. Estimations of local flashes for different Aν, adopted and Precision Physics Research located at Daejeon from Ref. [63]: the time duration and the corresponding event and COSINE-100 detector located at the Yangyang rate. Underground Laboratory in South Korea obviously satisfy the criteria dνa ≪ 0.1R⊕ when strongly correlated ampli- Aν τ (time span) Event rate fications for Aν and Aa may occur in both detectors almost 110s0.3 min−1 10 3 s 0.5 hr−1 12 102 1s 0.4 day−1 Daily modulations with an intensity around (1–10)% are 103 0.3 s 5 yr−1 similar in magnitude as annual modulations. These daily mod- 104 0.1 s 0.2 yr−1 ulations are unique for this type of DM and not shared by any other DM models; see [63] for the details. 083020-14 DAMA/LIBRA ANNUAL MODULATION AND AXION QUARK … PHYS. REV. D 101, 083020 (2020) simultaneously with a time delay of the order of a few must have a sharp cutoff at ∼6 keV consistent with the seconds. observed DL signal. We proposed specific tests which can Another correlation which is worthwhile to study can be support or rule our the proposal (1); see Sec. VC.We explained as follows. It has been recently argued that the proposed to study a specific correlation between the AQN propagating in Earth’s atmosphere and underground amplification in the AQN-induced neutron flux and the emits infrasound and weak seismic waves [87]. In fact, one impulses of the infrasound and weak seismic waves, which, such event, according to Ref. [87], occurred on 31 July, if found, would strongly support our proposal; see 2008. It was properly recorded by the dedicated Elginfield Sec. VD. Infrasound Array (ELFO) near London, Ontario, Canada, Why should we consider this AQN model seriously? and corresponds to a relatively large nugget13 with B ≃ 1027 There are a number of reasons. Originally, this model was Ω ∼ Ω if interpreted as an AQN event [87]. The infrasound invented to explain the observed relation DM visible and detection was accompanied by nonobservation of any the baryon asymmetry of the Universe as two sides of the meteors by an all-sky camera network. The impulses were same coin, when the baryogenesis framework is replaced also observed seismically as ground coupled acoustic waves by a “charge separation” framework, as reviewed in Sec. II. around southwestern Ontario and northern Michigan. The After many years since its original formulation, this model estimates [87] for the infrasonic frequency ν ≃ 5 Hz and remains consistent with all available cosmological, astro- overpressure δp ∼ 0.3 Pa are consistent with the ELFO physical, satellite, and ground-based constraints, where record. A detection strategy has been also proposed in AQNs could leave a detectable electromagnetic signature. Ref. [87] for a systematic study to search for such events Furthermore, it is shown that AQNs can be formed and can originating from much smaller and much more common survive the unfriendly environment during the evolution of AQNs with typical B ≃ 1025 by using distributed acoustic the early Universe, such that they are entitled to serve as sensing (DAS). DM candidates. Finally, the same AQN framework may Our original remark here is that the amplification in the also explain a number of other (naively unrelated) observed neutrino flux (and corresponding enhancement in the phenomena such as the excess of galactic diffuse emission neutrino-induced neutrons) as discussed in Sec. VC must in different frequency bands, the so-called primordial be accompanied by infrasonic and weak seismic waves lithium puzzle and the Solar corona mystery, and the which can be studied by the DAS instruments implemented seasonal variations observed by XMM-Newton observa- in networks of optical-fiber telecommunication cables. The tory, to name just a few; see Sec. I for the references. observation for such correlations, if successful, is abso- We want to emphasize that all these cosmological lutely unique to the AQN framework and would obviously puzzles mentioned in Sec. I could be resolved within the play an important role in supporting of the proposal. AQN framework with the same set of physical parameters In particular, DL (and future CYGNO detector) is being used in the present work on the explanation of the DL located at the LNGS site with a large number of seismic modulation signal. detectors located in the same area. The proposal is to search The observation of the subdominant AQN-induced for the correlations between the enhanced flux of neutrons neutrons by measuring the directionality and modulation and weak seismic and infrasound events with the delay as discussed in Sec. VBwould be a direct manifestation of measured in a fraction of a second depending on a precise the AQN dark matter model. The observation of a variety of localization of the seismic detectors. The measurements of amplifications as discussed in Sec. VC would be also a the directionality by CYGNO would be the key element in strong support for the proposal (1). Finally, the recording of establishing such a correlation. the correlations between the AQN-induced neutrino ampli- fication and the impulses of the infrasound and weak VI. CONCLUDING COMMENTS seismic waves would strongly support our proposal as argued in Sec. VD. We finish this work on this positive and The main results of our work can be summarized as optimistic note. follows. We argued that the annual modulation observed by DL ACKNOWLEDGMENTS might be explained as a result of the AQN-induced neutrons through the chain (1). In this framework, the annual I am thankful to G. Adhikari for answering my questions modulation has a truly genuine DM origin, though it is on the present status and future plans for the COSINE-100 manifested indirectly. We also argued that the recoil energy Collaboration (see footnote 10) and Elisabetta Baracchini for elaboration on future plans for the CYGNO Collabo- 13 ration (see footnote 11) during the PATRAS-2019 axion The frequency of appearance for such large nuggets is very meeting in Freiburg. I am thankful to Maria Martinez tiny according to Eq. (6). This explains why such events occur approximately once every 10 yr instead of observations of them [ANAIS-112] and Elisabetta Baracchini [CYGNO] for once a day, which would be the case for much more common correspondence. I am also thankful to Hyunsu Lee [CO- nuggets with B ∼ 1025. SINE] and Maria Luisa Sarsa [ANAIS-112] for discussions 083020-15 ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) during the “Conference on Dark World” in Daejeon, Korea, key characteristics are very much the same for all phases. where this work was presented. This research was sup- Therefore, we limit ourself below to reviewing the most ported in part by the Natural Sciences and Engineering developed, the so-called color flavor locking (CFL) phase. Research Council of Canada. The spontaneous breaking of chiral symmetry in color superconductors gives rise to low-energy pseudo-Nambu- APPENDIX A: NEUTRINO SPECTRUM FROM Goldstone (NG) modes with similar quantum numbers to THE AXION QUARK NUGGETS the mesons (pions, kaons, etc.). These objects, however, are collective excitations of the CS state rather than vacuum The main goal of this Appendix is to give a short excitations as is the case for the conventional confined overview of the basic results from Ref. [65] regarding the hadronic phase. The finite quark masses explicitly break neutrinos emitted by AQNs captured by the Sun. The paper chiral symmetry, giving rise to these “pseudo-Nambu- [65] was written in response to the claim made in Ref. [64] Goldstone” modes on the order of 20 MeV, in huge that dark matter in the form of AQNs cannot account contrast with the hadronic confined phase where the for more than 20% of the dark matter density. This lightest mass meson has mπ ≃ 140 MeV. claim was based on constraints on the neutrino flux in To be more precise, we consider a large μ limit for which the (20–50) MeV range, where the sensitivity of under- the masses and other relevant parameters in the CFL phase ground neutrino detectors such as Super-Kamiokande have can be explicitly computed [66,67]: their highest signal-to-noise ratio. 2c¯ However, the estimates [64] were based on an m2 ≃ m ðm þ m Þ;f2 ∼ μ2; π 2 s u d π assumption that the annihilation processes between anti- fπ matter from AQNs and normal material from the Sun have 2 2 2c¯ 3Δ the same spectral features as conventional baryon-anti- m ≃ m ðm þ m Þ; c¯ ≃ ; K f2 d u s 2π2 baryon annihilations, which typically produce a large π 2c¯ number of pions which eventually decay though an m2 ≃ m ðm þ m Þ: ð Þ K0 2 u d s A1 intermediate muon and, thus, generate a significant number fπ of neutrinos and antineutrinos in the (20–50) MeV range. However, as it has been argued in Ref. [65], the critical As one can see from Eq. (A1), the NG bosons are much difference in the case of annihilation processes involving an lighter than in a vacuum. This is because their masses are 2 antiquark nugget is that the annihilation proceeds within proportional to mq rather than to mq, as at zero chemical the CS phase, where the energetics are drastically different. potential. As a result, the lightest NG meson, the kaon, has The main point is that in most CS phases the lightest a mass in the range of 10–20 MeV depending on the precise pseudo-Goldstone mesons (the pions and kaons) have value of Δ and μ MeV; see, for example, [67]. masses in the 20 MeV range [66,67], in huge contrast Another important difference between the NG modes in with the hadronic confined phase where mπ ∼ 140 MeV. dense matter and in a vacuum is in the dispersion relations As a result of this crucial difference, the decay of light for the NG modes, which assume the following form (see, pseudo-Goldstone mesons of the CS phase cannot produce for example, [66]): neutrinos in the 20–50 MeV energy range and are not 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi subject to the SuperK constraints employed in Ref. [64]. ms 2 2 2 E ¼∓ þ v p þ m ; Instead, the pseudo-Goldstone mesons of the CS phase K 2μ NG K produce neutrinos in the 15 MeV range. 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ms 2 2 2 These unique spectral features of the neutrinos emitted E 0 ¼ − þ v p þ m ; K 2μ NG K0 by AQNs play the key role in our proposal, suggesting that 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the observed cutoff in DL modulations at 6 keV is ms 2 2 2 E 0 ¼þ þ v p þ m ; ultimately related to the cutoff in the neutrino energies K¯ 2μ NG K0 at 15 MeV emitted by AQNs. The emergence of this new qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 c 15 MeV energy scale is the subject of the next subsections. E ¼ v p þ m ;v¼ ; ð Þ π NG π NG 3 A2 1. Nambu-Goldstone modes in the CS phase such that the rest energy of the lightest NG mesons does not There are many possible CS phases due to the generation exceed the 10–20 MeV range. In fact, EK0 may even vanish, of a gap Δ through different channels with slightly different in which case the K0 field forms a condensate (the so-called K0 v properties. While the relevant physics is a part of the CFL phase). In these formulas, NG deviates from the standard model, QCD with no free parameters, the corre- speed of light c due to the explicit violation of the Lorentz sponding phase diagram is still a matter of debate, as it invariance in the system such that the dispersion relations strongly depends on the precise numerical value of the gap for all quasiparticles are drastically different from their Δ; see review articles [66,67]. For our purposes, though, the counterparts in a vacuum. 083020-16 DAMA/LIBRA ANNUAL MODULATION AND AXION QUARK … PHYS. REV. D 101, 083020 (2020) One should comment here that the dispersion relations (4) Larger chemical potentials μ generally lead to even for the NG modes within the antinuggets (which is most lighter NG modes, while the masses increase with relevant for our purposes) can be obtained from Eq. (A2) by the size of the gap Δ as Eqs. (A1) and (A2) state. replacing μ → −μ such that the lightest NG states become We conclude this subsection with the following generic the K¯ 0 and K− for nuggets made of antimatter. This comment. The main goal of Ref. [65] was to argue (in a ’ comment is important for identification of the neutrino simplified setting) that the neutrino s energies in the AQN and antineutrino spectra to be discussed in Appendix A2. framework are typically in the 15 MeV range (in contrast with the 20–50 MeV range as assumed in Ref. [64]) such that the stringent constraint from SuperK [69] does not apply. 2. Neutrino emission from NG bosons The neutrino emission from CFL phase quark matter has 3. Fermion excitations and neutrino’s been studied previously in a number of papers mostly in the production in CS phases context of the physics of neutron stars; see the original papers [88–90] and the review article [66]. As we already mentioned, the neutrino emission from CS In this subsection, we will overview the basic results of phase quark matter has been studied previously in a number Ref. [65] on neutrino and antineutrino fluxes from the Sun. of papers mostly in the context of the physics of neutron The main goal of Ref. [65] was to argue that the Super- stars; see the review article [66]. We cannot literally apply the machinery developed in previous studies, because all Kamiokande stringent constraint on antineutrino flux Φν¯ < e excitations (such as NG bosons) in our case are produced ð1.4–1.9Þ cm−2 s−1 at large energies E>19.3 MeV [69] not in a thermally equilibrium system when the density of (which played a key role in the analysis of Ref. [64])does the excitations is unambiguously determined by the temper- not apply to our case on neutrino and antineutrino produc- ature. Instead, all excitations in our scenario are produced tion by AQNs, because the typical energies of neutrinos and as a result of rare annihilation events. These annihilation antineutrinos will be much lower. events excite the NG bosons as well as fermion excitations, Indeed, the muons cannot be produced at all in the CFL which may decay by emitting neutrinos. The coefficients κν phase for purely kinematical reasons. Therefore, the ener- and κν¯ entering Eq. (16) precisely correspond to this getic antineutrinos which are normally produced in the μ → e ν ν mechanism of the neutrino production. e μ decay channels are not produced in the CS In this subsection, we want to overview some features of matter. This is the crucial point of the arguments presented the fermion excitations of the CS phases which may be the in Ref. [65]. dominant contributors to the neutrino fluxes. This is In the simplest possible scenario (which was adopted in because the NG bosons which are produced as a result Ref. [65]), the majority of neutrinos will be emitted by the of annihilation events in the CS phase cannot leave the lightest NG bosons, in which case the energy of the emitted system and consequently decay to emit neutrino, as they neutrinos does not normally exceed 15 MeV for the CFL must stay inside of the nuggets. It should be contrasted with phase, because the lightest NG bosons do not normally the hadronic phase when pions and kaons (produced as a exceed a mass in the 30 MeV range as mentioned in result of, e.g., pp¯ annihilation) decay to muons and Appendix A1. This is a very basic and very generic feature neutrinos in a vacuum. Therefore, the NG bosons in CS of the CS phase. We postpone the important discussions on phases are likely to be absorbed by fermion excitations (if basic features of the neutrinos emitted by the quarks in CS kinematically allowed), which consequently decay to phases to Appendix A3. Below, we list a few features on neutrinos. the neutrino spectrum if it is saturated exclusively by NG We start our overview by mentioning some CS phases bosons: which support low-energy fermion excitations. Detailed μ Δ (1) A specific choice of and determines the basic discussions can be found in Ref. [66]. First of all, the so- mass scales for the light NG modes, which even- called 2SC phase (when two out of three colors and flavors ν tually determine the spectrum. are paired and condensed) supports unpaired modes which μ (2) It is normally assumed that a required value for for could be light and couple to NG bosons. Another phase μ ≃ 400 CFL to be realized is MeV. The corre- which also supports the light fermion excitations is the so- Δ sponding value for is estimated in this case as called CFL-K0 phase when K0 energy vanishes according Δ ≃ 100 MeV in the CFL phase; see Ref. [66]. Such 0 to Eq. (A2). For antinuggets, this phase corresponds to K¯ a large value of μ can be indeed reached in the condensation. In both cases (CFL and CFL-K0), the gap of AQN’s core as recent numerical studies suggest; see the excitations decreases with increases of pF such that the Fig. 2 in Ref. [46]. 2 ms gaps for p and n become lower. Indeed, Δp;n ¼ Δ − (3) The neutrino spectrum is qualitatively different from 2pF that of the antineutrinos, because the annihilation such that p, n fermion excitations could become com- occurs not in a vacuum but in a dense CS state pletely ungapped. If this happens, these modes may with μ ≃ 400 MeV. become the dominant producers of the neutrinos. 083020-17 ARIEL ZHITNITSKY PHYS. REV. D 101, 083020 (2020) β z Indeed, in unpaired quark matter, neutrino emissivity where is the angle related to cut as shown in Fig. 1, and is dominated by the direct Urca processes such as we used formula (13) which defines the coefficient κν as the − d → u þ e þ ν¯e. In the case of antinuggets, it should number of neutrinos produced due to the annihilation of a be replaced by antiquarks with emission of neutrino νe and single baryon charge. In obtaining Eq. (B1), the integration eþ z m positron with the energy determined by the energy of the d is replaced by integration d AQN: fermion excitation, which itself assumes the energy of the κ κ m_ order of the lightest NG mode according to analysis of N ≃ ν m ¼ ν AQN z: ð Þ d ν m d AQN m v d B2 Appendix A2. p p AQN If this process indeed becomes the dominant mechanism of the neutrino emission from AQNs, then one should Now we can estimate the flux of neutrinos due to the expect that passage of this nearby AQN as follows: dNν 1 dNν κν β κν ≫ κν¯ ;Eν ≲ 15 MeV; κν ≳ 1; ðA3Þ Δ ≃ Δ ¼ m_ ; ð Þ 2 AQN B3 dAdt τ dA mp 4πd which is assumed to be the case as Eq. (13) states. where we used expression (B1) and approximated τ ≃ 2d=v AQN as a typical travel time for an AQN inside the APPENDIX B: LOCAL FLASHES ½−z ;z interval cut cut . In this Appendix, we generalize our axion studies [63] to We want to compare this local flash (B3) with the include the neutrinos into consideration, similar to what we average flux (14) by introducing the amplification factor have done in Sec. III B. The main topic for the present AνðdÞ defined as the ratio: studies is the enhancement effect and great amplification of the axion density which was coined as a “local flash” in Δð dNν Þ β R 2 A ðdÞ ≡ dAdt ≃ ⊕ ; ð Þ Ref. [63]. It occurs on rare occasions when an AQN hits (or ν N B4 ð d ν Þ hN_ ihΔti d exits) Earth’s surface in the vicinity of an axion search dAdt detector. In this Appendix, we generalize the arguments of m_ ≃ hΔm i=hΔti Sec. III B to estimate a similar local flash for neutrinos. where we approximated AQN AQN . The hΔti We follow the same logic of Ref. [63] and consider a case physical meaning of is the time duration of the when an AQN is moving in a distance d close enough to the AQNs being averaged over all trajectories and over the detector, as shown in Fig. 1. The total number of emitted velocity distribution. The result (B4) does not depend on ’ neutrinos per unit area within z ∈ ½−z ;z as a result of the neutrino s spectrum nor intensity. It does not include cut cut κ passage of the AQN at distance d from an observer is even parameter ν and, in fact, identically coincides with given by the expression obtained for the axion local flash derived Z previously in Ref. [63]. It is an anticipated result, as all N 1 z κ m ðzÞ Δ d ν ¼ cut ν d AQN these numerical factors cancel out in the ratio (B4) as A 4π z2 þ d2 m relative amplification factor (B4) is entirely determined by d −zcut p β κ m_ the dynamics of the AQNs, not the particles they emit as ≃ ν AQN ; ð Þ long as these particles are relativistic, which is the case for 2πd v m B1 AQN p both species: the axions and neutrinos. As the final expression (B4) for the neutrino amplifica- tion factor coincides with the corresponding expression for the axion [63], the consequences in both cases are the same, and we simply list them. (1) For the typical ratio hN_ ihΔti ∼ 30, where hN_ i is given by Eq. (9) and β ∼ 1, one can infer that an amplification becomes significant if d ≪ 0.1R⊕. (2) The time duration τ of a local flash as a function of amplification Aν: 2d hΔti 1=2 1 τ ≃ ≃ ; ð Þ v _ 1=2 B5 AQN hNi Aν v ≃ where, in the last step, we approximate AQN 2R⊕=hΔti and assume β ∼ 1 for simplicity. We FIG. 1. 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