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Dequantization of electric charge: probing scenarios of cosmological multi-component dark matter. (English) Zbl 1507.83044

Summary: Since the electric charge in the standard model is theoretically not quantized, we may have a variant of it, called dark charge. Similar to the electric charge, the dark charge neither commutes nor closes algebraically with \(SU(2)_L\). The condition of algebraic closure leads to a novel gauge extension, \(SU(2)_L \otimes U(1)_Y \otimes U(1)_N\), where \(Y\) and \(N\) determine the electric and dark charges, respectively, apart from the color group. We argue that the existence of the dark charge, thus \(N\), leads to novel scenarios of multi-component dark matter, in general. The dark matter stability is determined by a residual (or dark charge) gauge symmetry isomorphic to an even \(Z_k\) discrete group, where \(k\) is specified dependent on the value of the neutrino dark charge. This residual symmetry divides the standard model particles into distinct classes, which possibly accommodate dark matter, but each dark matter candidate cannot decay due to the color and electric charge conservation. We analyze in detail three specific models according to \(k = 2, 4, 6\) and determine the simplest dark matter candidates. For small \(U(1)_N\) coupling, the two-component dark matter scenarios implied by the dark charge successfully explain the dark matter relic density and the recent XENON1T excess, as well as the beam dump, neutrino scattering, and astrophysical bounds. Otherwise, for large \(U(1)_N\) coupling, we have multi-WIMPs coexisted beyond the weak scale.

MSC:

83C56 Dark matter and dark energy
83F05 Relativistic cosmology
81V22 Unified quantum theories
78A35 Motion of charged particles
51F25 Orthogonal and unitary groups in metric geometry
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
17B75 Color Lie (super)algebras
22E40 Discrete subgroups of Lie groups
81V15 Weak interaction in quantum theory

Keywords:

dark charge

References:

[1] Review of particle physics, PTEP, 2020, Article 083C01 pp. (2020)
[2] Kajita, T., Nobel lecture: discovery of atmospheric neutrino oscillations, Rev. Mod. Phys., 88, Article 030501 pp. (2016)
[3] McDonald, A. B., Nobel lecture: the Sudbury neutrino observatory: observation of flavor change for solar neutrinos, Rev. Mod. Phys., 88, Article 030502 pp. (2016)
[4] Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological parameter results, Astrophys. J. Suppl., 208, 19 (2013)
[5] Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys., 641, A6 (2020)
[6] Jungman, G.; Kamionkowski, M.; Griest, K., Supersymmetric dark matter, Phys. Rep., 267, 195 (1996)
[7] Bertone, G.; Hooper, D.; Silk, J., Particle dark matter: evidence, candidates and constraints, Phys. Rep., 405, 279 (2005)
[8] Berezhiani, Z. G.; Khlopov, M. Y., Cosmology of spontaneously broken gauge family symmetry, Z. Phys. C, 49, 73 (1991)
[9] Berezhiani, Z. G.; Khlopov, M. Y., Physics of cosmological dark matter in the theory of broken family symmetry, Sov. J. Nucl. Phys., 52, 60 (1990), (in Russian)
[10] Ma, E., Supersymmetric model of radiative seesaw Majorana neutrino masses, Ann. Fond. Broglie, 31, 285 (2006)
[11] Zurek, K. M., Multi-component dark matter, Phys. Rev. D, 79, Article 115002 pp. (2009)
[12] Fukuoka, H.; Suematsu, D.; Toma, T., Signals of dark matter in a supersymmetric two dark matter model, J. Cosmol. Astropart. Phys., 1107, Article 001 pp. (2011)
[13] Batell, B., Dark discrete gauge symmetries, Phys. Rev. D, 83, Article 035006 pp. (2011)
[14] Chialva, D.; Dev, P.; Mazumdar, A., Multiple dark matter scenarios from ubiquitous stringy throats, Phys. Rev. D, 87, Article 063522 pp. (2013)
[15] Biswas, A.; Majumdar, D.; Sil, A.; Bhattacharjee, P., Two component dark matter: a possible explanation of 130 GeV γ− ray line from the galactic centre, J. Cosmol. Astropart. Phys., 1312, Article 049 pp. (2013)
[16] Bhattacharya, S.; Drozd, A.; Grzadkowski, B.; Wudka, J., Two-component dark matter, J. High Energy Phys., 10, Article 158 pp. (2013)
[17] Bian, L.; Ding, R.; Zhu, B., Two component Higgs-portal dark matter, Phys. Lett. B, 728, 105 (2014) · Zbl 1377.81247
[18] Bélanger, G.; Kannike, K.; Pukhov, A.; Raidal, M., Minimal semi-annihilating \(Z_N\) scalar dark matter, J. Cosmol. Astropart. Phys., 06, Article 021 pp. (2014)
[19] Bian, L.; Li, T.; Shu, J.; Wang, X.-C., Two component dark matter with multi-Higgs portals, J. High Energy Phys., 03, Article 126 pp. (2015)
[20] Esch, S.; Klasen, M.; Yaguna, C. E., A minimal model for two-component dark matter, J. High Energy Phys., 09, Article 108 pp. (2014)
[21] DiFranzo, A.; Mohlabeng, G., Multi-component dark matter through a radiative Higgs portal, J. High Energy Phys., 01, Article 080 pp. (2017) · Zbl 1373.85008
[22] Dutta Banik, A.; Pandey, M.; Majumdar, D.; Biswas, A., Two component WIMP-FImP dark matter model with singlet fermion, scalar and pseudo scalar, Eur. Phys. J. C, 77, 657 (2017)
[23] Karam, A.; Tamvakis, K., Dark matter from a classically scale-invariant \(S U ( 3 )_X\), Phys. Rev. D, 94, Article 055004 pp. (2016)
[24] Bhattacharya, S.; Poulose, P.; Ghosh, P., Multipartite interacting scalar dark matter in the light of updated LUX data, J. Cosmol. Astropart. Phys., 04, Article 043 pp. (2017)
[25] Arcadi, G.; Gross, C.; Lebedev, O.; Mambrini, Y.; Pokorski, S.; Toma, T., Multicomponent dark matter from gauge symmetry, J. High Energy Phys., 12, Article 081 pp. (2016) · Zbl 1390.83431
[26] Borah, D.; Dasgupta, A.; Dey, U. K.; Patra, S.; Tomar, G., Multi-component fermionic dark matter and IceCube PeV scale neutrinos in left-right model with gauge unification, J. High Energy Phys., 09, Article 005 pp. (2017)
[27] Ahmed, A.; Duch, M.; Grzadkowski, B.; Iglicki, M., Multi-component dark matter: the vector and fermion case, Eur. Phys. J. C, 78, 905 (2018)
[28] Bhattacharya, S.; Ghosh, P.; Maity, T. N.; Ray, T. S., Mitigating direct detection bounds in non-minimal Higgs portal scalar dark matter models, J. High Energy Phys., 10, Article 088 pp. (2017)
[29] Bhattacharya, S.; Ghosh, P.; Sahu, N., Multipartite dark matter with scalars, fermions and signatures at LHC, J. High Energy Phys., 02, Article 059 pp. (2019)
[30] Bhattacharya, S.; Saha, A. K.; Sil, A.; Wudka, J., Dark matter as a remnant of SQCD inflation, J. High Energy Phys., 10, Article 124 pp. (2018) · Zbl 1402.85006
[31] Aoki, M.; Toma, T., Boosted self-interacting dark matter in a multi-component dark matter model, J. Cosmol. Astropart. Phys., 1810, Article 020 pp. (2018)
[32] Dutta Banik, A.; Saha, A. K.; Sil, A., Scalar assisted singlet doublet fermion dark matter model and electroweak vacuum stability, Phys. Rev. D, 98, Article 075013 pp. (2018)
[33] Barman, B.; Bhattacharya, S.; Zakeri, M., Multipartite dark matter in \(S U ( 2 )_N\) extension of standard model and signatures at the LHC, J. Cosmol. Astropart. Phys., 1809, Article 023 pp. (2018)
[34] Ayazi, S. Yaser; Mohamadnejad, A., Scale-invariant two component dark matter, Eur. Phys. J. C, 79, 140 (2019)
[35] Chakraborti, S.; Dutta Banik, A.; Islam, R., Probing multicomponent extension of inert doublet model with a vector dark matter, Eur. Phys. J. C, 79, 662 (2019)
[36] Chakraborti, S.; Poulose, P., Interplay of scalar and fermionic components in a multi-component dark matter scenario, Eur. Phys. J. C, 79, 420 (2019)
[37] Borah, D.; Dasgupta, A.; Kang, S. K., Two-component dark matter with cogenesis of the baryon asymmetry of the Universe, Phys. Rev. D, 100, Article 103502 pp. (2019)
[38] Elahi, F.; Khatibi, S., Multi-component dark matter in a non-Abelian dark sector, Phys. Rev. D, 100, Article 015019 pp. (2019)
[39] Yaguna, C. E.; Zapata, O., Multi-component scalar dark matter from a \(Z_N\) symmetry: a systematic analysis, J. High Energy Phys., 03, Article 109 pp. (2020)
[40] Fan, J.; Katz, A.; Randall, L.; Reece, M., Double-disk dark matter, Phys. Dark Universe, 2, 139 (2013)
[41] Fan, J.; Katz, A.; Randall, L.; Reece, M., Dark-disk Universe, Phys. Rev. Lett., 110, Article 211302 pp. (2013)
[42] Boehm, C.; Fayet, P.; Silk, J., Light and heavy dark matter particles, Phys. Rev. D, 69, Article 101302 pp. (2004)
[43] Agashe, K.; Cui, Y.; Necib, L.; Thaler, J., (In)direct detection of boosted dark matter, J. Cosmol. Astropart. Phys., 10, Article 062 pp. (2014)
[44] Kong, K.; Mohlabeng, G.; Park, J.-C., Boosted dark matter signals uplifted with self-interaction, Phys. Lett. B, 743, 256 (2015)
[45] Alhazmi, H.; Kong, K.; Mohlabeng, G.; Park, J.-C., Boosted dark matter at the deep underground neutrino experiment, J. High Energy Phys., 04, Article 158 pp. (2017)
[46] Kim, D.; Park, J.-C.; Shin, S., Dark matter “collider” from inelastic boosted dark matter, Phys. Rev. Lett., 119, Article 161801 pp. (2017)
[47] Giudice, G. F.; Kim, D.; Park, J.-C.; Shin, S., Inelastic boosted dark matter at direct detection experiments, Phys. Lett. B, 780, 543 (2018)
[48] Chatterjee, A.; De Roeck, A.; Kim, D.; Moghaddam, Z. G.; Park, J.-C.; Shin, S., Searching for boosted dark matter at ProtoDUNE, Phys. Rev. D, 98, Article 075027 pp. (2018)
[49] Kim, D.; Kong, K.; Park, J.-C.; Shin, S., Boosted dark matter quarrying at surface neutrino detectors, J. High Energy Phys., 08, Article 155 pp. (2018)
[50] Elbert, O. D.; Bullock, J. S.; Garrison-Kimmel, S.; Rocha, M.; Oñorbe, J.; Peter, A. H., Core formation in dwarf haloes with self-interacting dark matter: no fine-tuning necessary, Mon. Not. R. Astron. Soc., 453, 29 (2015)
[51] Tulin, S.; Yu, H.-B., Dark matter self-interactions and small scale structure, Phys. Rep., 730, 1 (2018) · Zbl 1381.83158
[52] Heeck, J.; Zhang, H., Exotic charges, multicomponent dark matter and light sterile neutrinos, J. High Energy Phys., 05, Article 164 pp. (2013) · Zbl 1342.81681
[53] Aoki, M.; Duerr, M.; Kubo, J.; Takano, H., Multi-component dark matter systems and their observation prospects, Phys. Rev. D, 86, Article 076015 pp. (2012)
[54] Aoki, M.; Kubo, J.; Takano, H., Two-loop radiative seesaw mechanism with multicomponent dark matter explaining the possible γ excess in the Higgs boson decay and at the Fermi LAT, Phys. Rev. D, 87, Article 116001 pp. (2013)
[55] Kajiyama, Y.; Okada, H.; Toma, T., Multicomponent dark matter particles in a two-loop neutrino model, Phys. Rev. D, 88, Article 015029 pp. (2013)
[56] Karam, A.; Tamvakis, K., Dark matter and neutrino masses from a scale-invariant multi-Higgs portal, Phys. Rev. D, 92, Article 075010 pp. (2015)
[57] Bernal, N.; Restrepo, D.; Yaguna, C.; Zapata, O., Two-component dark matter and a massless neutrino in a new \(B - L\) model, Phys. Rev. D, 99, Article 015038 pp. (2019)
[58] Bonilla, C.; Centelles-Chuliá, S.; Cepedello, R.; Peinado, E.; Srivastava, R., Dark matter stability and Dirac neutrinos using only Standard Model symmetries, Phys. Rev. D, 101, Article 033011 pp. (2020)
[59] Borah, D.; Roshan, R.; Sil, A., Minimal two-component scalar doublet dark matter with radiative neutrino mass, Phys. Rev. D, 100, Article 055027 pp. (2019)
[60] Bhattacharya, S.; Ghosh, P.; Saha, A. K.; Sil, A., Two component dark matter with inert Higgs doublet: neutrino mass, high scale validity and collider searches
[61] Biswas, A.; Borah, D.; Nanda, D., Type III seesaw for neutrino masses in \(U(1){}_{B - L}\) model with multi-component dark matter, J. High Energy Phys., 12, Article 109 pp. (2019)
[62] Bhattacharya, S.; Chakrabarty, N.; Roshan, R.; Sil, A., Multicomponent dark matter in extended \(U ( 1 )_{B - L}\): neutrino mass and high scale validity, J. Cosmol. Astropart. Phys., 04, Article 013 pp. (2020)
[63] Van Dong, P., Flipping principle for neutrino mass and dark matter, Phys. Rev. D, 102, Article 011701 pp. (2020)
[64] Van Loi, D.; Nam, C. H.; Tan, N. H.; Van Dong, P., Dark charge vs electric charge
[65] Dong, P.; Hung, H.; Tham, T., 3-3-1-1 model for dark matter, Phys. Rev. D, 87, Article 115003 pp. (2013)
[66] Dong, P. V., Unifying the electroweak and B-L interactions, Phys. Rev. D, 92, Article 055026 pp. (2015)
[67] Alves, A.; Arcadi, G.; Dong, P.; Duarte, L.; Queiroz, F. S.; Valle, J. W.F., Matter-parity as a residual gauge symmetry: probing a theory of cosmological dark matter, Phys. Lett. B, 772, 825 (2017)
[68] Dong, P.; Huong, D.; Loi, D.; Nhuan, N.; Ngan, N., Phenomenology of the \(S U ( 3 )_C \otimes S U ( 2 )_L \otimes S U ( 3 )_R \otimes U ( 1 )_X\) gauge model, Phys. Rev. D, 95, Article 075034 pp. (2017)
[69] Dong, P. V.; Huong, D. T.; Queiroz, F. S.; Valle, J. W.F.; Vaquera-Araujo, C. A., The dark side of flipped trinification, J. High Energy Phys., 04, Article 143 pp. (2018)
[70] Van Dong, P.; Huong, D. T.; Camargo, D. A.; Queiroz, F. S.; Valle, J. W.F., Asymmetric dark matter, inflation and leptogenesis from \(B - L\) symmetry breaking, Phys. Rev. D, 99, Article 055040 pp. (2019)
[71] Nam, C. H.; Van Loi, D.; Thuy, L. X.; Van Dong, P., Multicomponent dark matter in noncommutative \(B - L\) gauge theory, J. High Energy Phys., 12, Article 029 pp. (2020)
[72] Van Dong, P.; Nam, C. H.; Van Loi, D., Canonical seesaw implication for two-component dark matter, Phys. Rev. D, 103, Article 095016 pp. (2021)
[73] Van Loi, D.; Van Dong, P.; Van Soa, D., Neutrino mass and dark matter from an approximate B − L symmetry, J. High Energy Phys., 05, Article 090 pp. (2020)
[74] Van Loi, D.; Nam, C. H.; Van Dong, P., Dark matter in the fully flipped 3-3-1-1 model
[75] Dirac, P. A.M., Quantised singularities in the electromagnetic field, Proc. R. Soc. Lond. A, 133, 60 (1931) · Zbl 0002.30502
[76] Babu, K. S.; Mohapatra, R. N., Quantization of electric charge from anomaly constraints and a Majorana neutrino, Phys. Rev. D, 41, 271 (1990)
[77] Foot, R.; Joshi, G. C.; Lew, H.; Volkas, R. R., Charge quantization in the standard model and some of its extensions, Mod. Phys. Lett. A, 5, 2721 (1990)
[78] Pisano, F., A simple solution for the flavor question, Mod. Phys. Lett. A, 11, 2639 (1996)
[79] Doff, A.; Pisano, F., Charge quantization in the largest leptoquark bilepton chiral electroweak scheme, Mod. Phys. Lett. A, 14, 1133 (1999)
[80] de Sousa Pires, C. A.; Ravinez, O. P., Charge quantization in a chiral bilepton gauge model, Phys. Rev. D, 58, Article 035008 pp. (1998)
[81] de Sousa Pires, C. A., Remark on the vector - like nature of the electromagnetism and the electric charge quantization, Phys. Rev. D, 60, Article 075013 pp. (1999)
[82] Dong, P. V.; Long, H. N., Electric charge quantization in SU(3) (C) x SU(3) (L) x U(1) (X) models, Int. J. Mod. Phys. A, 21, 6677 (2006) · Zbl 1107.81344
[83] Fermi LAT search for dark matter in gamma-ray lines and the inclusive photon spectrum, Phys. Rev. D, 86, Article 022002 pp. (2012)
[84] Appelquist, T.; Dobrescu, B. A.; Hopper, A. R., Nonexotic neutral gauge bosons, Phys. Rev. D, 68, Article 035012 pp. (2003)
[85] Carena, M.; Daleo, A.; Dobrescu, B. A.; Tait, T. M.P., \( Z^\prime\) gauge bosons at the tevatron, Phys. Rev. D, 70, Article 093009 pp. (2004)
[86] Search for new high-mass phenomena in the dilepton final state using 36 fb^−1 of proton-proton collision data at \(\sqrt{ s} = 13\) TeV with the ATLAS detector, J. High Energy Phys., 10, Article 182 pp. (2017)
[87] Faraggi, A. E.; Mehta, V. M., Proton stability and light \(Z^\prime\) inspired by string derived models, Phys. Rev. D, 84, Article 086006 pp. (2011)
[88] Huong, D. T.; Dong, P. V., Left-right asymmetry and 750 GeV diphoton excess, Phys. Rev. D, 93, Article 095019 pp. (2016)
[89] Excess electronic recoil events in XENON1T, Phys. Rev. D, 102, Article 072004 pp. (2020)
[90] Kannike, K.; Raidal, M.; Veermäe, H.; Strumia, A.; Teresi, D., Dark matter and the XENON1T electron recoil excess, Phys. Rev. D, 102, Article 095002 pp. (2020)
[91] Fornal, B.; Sandick, P.; Shu, J.; Su, M.; Zhao, Y., Boosted dark matter interpretation of the XENON1T excess, Phys. Rev. Lett., 125, Article 161804 pp. (2020)
[92] Primulando, R.; Julio, J.; Uttayarat, P., Collider constraints on a dark matter interpretation of the XENON1T excess, Eur. Phys. J. C, 80, 1084 (2020)
[93] Su, L.; Wang, W.; Wu, L.; Yang, J. M.; Zhu, B., Atmospheric dark matter and Xenon1T excess, Phys. Rev. D, 102, Article 115028 pp. (2020)
[94] Cao, Q.-H.; Ding, R.; Xiang, Q.-F., Searching for sub-MeV boosted dark matter from xenon electron direct detection, Chin. Phys. C, 45, Article 045002 pp. (2021)
[95] Alhazmi, H.; Kim, D.; Kong, K.; Mohlabeng, G.; Park, J.-C.; Shin, S., Implications of the XENON1T excess on the dark matter interpretation, J. High Energy Phys., 05, Article 055 pp. (2021)
[96] Delle Rose, L.; Hütsi, G.; Marzo, C.; Marzola, L., Impact of loop-induced processes on the boosted dark matter interpretation of the XENON1T excess, J. Cosmol. Astropart. Phys., 02, Article 031 pp. (2021)
[97] Ko, P.; Tang, Y., Semi-annihilating \(Z_3\) dark matter for XENON1T excess, Phys. Lett. B, 815, Article 136181 pp. (2021)
[98] Dey, U. K.; Maity, T. N.; Ray, T. S., Prospects of migdal effect in the explanation of XENON1T electron recoil excess, Phys. Lett. B, 811, Article 135900 pp. (2020)
[99] Sabti, N.; Alvey, J.; Escudero, M.; Fairbairn, M.; Blas, D., Refined bounds on MeV-scale thermal dark sectors from BBN and the CMB, J. Cosmol. Astropart. Phys., 01, Article 004 pp. (2020)
[100] Joglekar, A.; Raj, N.; Tanedo, P.; Yu, H.-B., Relativistic capture of dark matter by electrons in neutron stars, Phys. Lett. B, Article 135767 pp. (2020)
[101] Navarro, J. F.; Frenk, C. S.; White, S. D.M., The structure of cold dark matter halos, Astrophys. J., 462, 563 (1996)
[102] Ilten, P.; Soreq, Y.; Williams, M.; Xue, W., Serendipity in dark photon searches, J. High Energy Phys., 06, Article 004 pp. (2018)
[103] Bauer, M.; Foldenauer, P.; Jaeckel, J., Hunting all the hidden photons, J. High Energy Phys., 07, Article 094 pp. (2018)
[104] Kawasaki, M.; Steigman, G.; Kang, H.-S., Cosmological evolution of an early decaying particle, Nucl. Phys. B, 403, 671 (1993)
[105] Escudero, M.; Hooper, D.; Krnjaic, G.; Pierre, M., Cosmology with a very light L_μ − L_τ gauge boson, J. High Energy Phys., 03, Article 071 pp. (2019)
[106] Riess, A. G., A 2.4
[107] Dong, P. V.; Nguyen, T. P.; Soa, D. V., 3-3-1 model with inert scalar triplet, Phys. Rev. D, 88, Article 095014 pp. (2013)
[108] Dong, P. V.; Ngan, N. T.K.; Soa, D. V., Simple 3-3-1 model and implication for dark matter, Phys. Rev. D, 90, Article 075019 pp. (2014)
[109] Dong, P. V.; Huong, D. T.; Queiroz, F. S.; Thuy, N. T., Phenomenology of the 3-3-1-1 model, Phys. Rev. D, 90, Article 075021 pp. (2014)
[110] Dong, P. V.; Kim, C. S.; Soa, D. V.; Thuy, N. T., Investigation of dark matter in minimal 3-3-1 models, Phys. Rev. D, 91, Article 115019 pp. (2015)
[111] Van Dong, P.; Ngan, N. T.K.; Tham, T. D.; Thien, L. D.; Thuy, N. T., Phenomenology of the simple 3-3-1 model with inert scalars, Phys. Rev. D, 99, Article 095031 pp. (2019)
[112] Huong, D. T.; Dong, P. V., Neutrino masses and superheavy dark matter in the 3-3-1-1 model, Eur. Phys. J. C, 77, 204 (2017)
[113] Huong, D. T.; Dong, P. V.; Duy, N. T.; Nhuan, N. T.; Thien, L. D., Investigation of dark matter in the 3-2-3-1 model, Phys. Rev. D, 98, Article 055033 pp. (2018)
[114] Huong, D. T.; Dinh, D. N.; Thien, L. D.; Van Dong, P., Dark matter and flavor changing in the flipped 3-3-1 model, J. High Energy Phys., 08, Article 051 pp. (2019)
[115] Dinh, D. N.; Huong, D. T.; Duy, N. T.; Nhuan, N. T.; Thien, L. D.; Van Dong, P., Flavor changing in the flipped trinification, Phys. Rev. D, 99, Article 055005 pp. (2019)
[116] Dong, P. V.; Huong, D. T., Left-right model for dark matter, Commun. Phys., 28, 21 (2018)
[117] Dong, P. V.; Phong, D. Q.; Soa, D. V.; Thao, N. C., The economical 3-3-1 model revisited, Eur. Phys. J. C, 78, 653 (2018)
[118] Van Dong, P.; Van Loi, D., Asymmetric matter from \(B - L\) symmetry breaking, Eur. Phys. J. C, 80, 1137 (2020)
[119] First dark matter search results from the XENON1T experiment, Phys. Rev. Lett., 119, Article 181301 pp. (2017)
[120] Dark matter search results from a one ton-year exposure of XENON1T, Phys. Rev. Lett., 121, Article 111302 pp. (2018)
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