Abstract
The formation of meta-stable dark matter bound states in coannihilating scenarios could efficiently occur through the scattering with a variety of Standard Model bath particles, where light bosons during the electroweak cross over or even massless photons and gluons are exchanged in the t-channel. The amplitudes for those higher-order processes, however, are divergent in the collinear direction of the in- and out-going bath particles if the mediator is massless. To address the issue of collinear divergences, we derive the bound-state formation collision term in the framework of non-equilibrium quantum field theory. The main result is an expression for a more general cross section, which allows to compute higher-order bound-state formation processes inside the primordial plasma background in a comprehensive manner. Based on this result, we show that next-to-leading order contributions, including the bath-particle scattering, are i) collinear finite and ii) generically dominate over the on-shell emission for temperatures larger than the absolute value of the binding energy. Based on a simplified model, we demonstrate that the impact of these new effects on the thermal relic abundance is significant enough to make it worthwhile to study more realistic coannihilation scenarios.
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B.W. Lee and S. Weinberg, Cosmological lower bound on heavy neutrino masses, Phys. Rev. Lett. 39 (1977) 165 [INSPIRE].
J.R. Ellis, J.S. Hagelin, D.V. Nanopoulos, K.A. Olive and M. Srednicki, Supersymmetric relics from the big bang, Nucl. Phys. B 238 (1984) 453 [INSPIRE].
G. Arcadi et al., The waning of the WIMP? A review of models, searches and constraints, Eur. Phys. J. C 78 (2018) 203 [arXiv:1703.07364] [INSPIRE].
Planck collaboration, Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE].
J. Hisano, S. Matsumoto and M.M. Nojiri, Unitarity and higher order corrections in neutralino dark matter annihilation into two photons, Phys. Rev. D 67 (2003) 075014 [hep-ph/0212022] [INSPIRE].
J. Hisano, S. Matsumoto and M.M. Nojiri, Explosive dark matter annihilation, Phys. Rev. Lett. 92 (2004) 031303 [hep-ph/0307216] [INSPIRE].
J. Hisano, S. Matsumoto, M.M. Nojiri and O. Saito, Non-perturbative effect on dark matter annihilation and gamma ray signature from galactic center, Phys. Rev. D 71 (2005) 063528 [hep-ph/0412403] [INSPIRE].
J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Non-perturbative effect on thermal relic abundance of dark matter, Phys. Lett. B 646 (2007) 34 [hep-ph/0610249] [INSPIRE].
A. Sommerfeld, Über die Beugung und Bremsung der Elektronen (in German), Annalen Phys. 403 (1931) 257.
A.D. Sakharov, Interaction of an electron and positron in pair production, Sov. Phys. Usp. 34 (1991) 375 [Zh. Eksp. Teor. Fiz. 18 (1948) 631] [Usp. Fiz. Nauk 161 (1991) 29] [INSPIRE].
A. Freitas, Radiative corrections to co-annihilation processes, Phys. Lett. B 652 (2007) 280 [arXiv:0705.4027] [INSPIRE].
M. Cirelli, A. Strumia and M. Tamburini, Cosmology and astrophysics of minimal dark matter, Nucl. Phys. B 787 (2007) 152 [arXiv:0706.4071] [INSPIRE].
C.F. Berger, L. Covi, S. Kraml and F. Palorini, The number density of a charged relic, JCAP 10 (2008) 005 [arXiv:0807.0211] [INSPIRE].
M. Drees, J.M. Kim and K.I. Nagao, Potentially large one-loop corrections to WIMP annihilation, Phys. Rev. D 81 (2010) 105004 [arXiv:0911.3795] [INSPIRE].
A. Hryczuk, R. Iengo and P. Ullio, Relic densities including Sommerfeld enhancements in the MSSM, JHEP 03 (2011) 069 [arXiv:1010.2172] [INSPIRE].
A. Hryczuk, The Sommerfeld enhancement for scalar particles and application to sfermion co-annihilation regions, Phys. Lett. B 699 (2011) 271 [arXiv:1102.4295] [INSPIRE].
A. Hryczuk and R. Iengo, The one-loop and Sommerfeld electroweak corrections to the wino dark matter annihilation, JHEP 01 (2012) 163 [Erratum ibid. 06 (2012) 137] [arXiv:1111.2916] [INSPIRE].
M. Beneke, C. Hellmann and P. Ruiz-Femenia, Non-relativistic pair annihilation of nearly mass degenerate neutralinos and charginos I. General framework and S-wave annihilation, JHEP 03 (2013) 148 [Erratum ibid. 10 (2013) 224] [arXiv:1210.7928] [INSPIRE].
K. Harigaya, K. Kaneta and S. Matsumoto, Gaugino coannihilations, Phys. Rev. D 89 (2014) 115021 [arXiv:1403.0715] [INSPIRE].
J. Harz, B. Herrmann, M. Klasen, K. Kovařík and M. Meinecke, SUSY-QCD corrections to stop annihilation into electroweak final states including Coulomb enhancement effects, Phys. Rev. D 91 (2015) 034012 [arXiv:1410.8063] [INSPIRE].
M. Beneke, C. Hellmann and P. Ruiz-Femenia, Heavy neutralino relic abundance with Sommerfeld enhancements — a study of pMSSM scenarios, JHEP 03 (2015) 162 [arXiv:1411.6930] [INSPIRE].
J. Harz, B. Herrmann, M. Klasen, K. Kovařík and P. Steppeler, Theoretical uncertainty of the supersymmetric dark matter relic density from scheme and scale variations, Phys. Rev. D 93 (2016) 114023 [arXiv:1602.08103] [INSPIRE].
M. Beneke et al., Relic density of wino-like dark matter in the MSSM, JHEP 03 (2016) 119 [arXiv:1601.04718] [INSPIRE].
M. Beneke, A. Bharucha, A. Hryczuk, S. Recksiegel and P. Ruiz-Femenia, The last refuge of mixed wino-Higgsino dark matter, JHEP 01 (2017) 002 [arXiv:1611.00804] [INSPIRE].
S. El Hedri, A. Kaminska and M. de Vries, A Sommerfeld toolbox for colored dark sectors, Eur. Phys. J. C 77 (2017) 622 [arXiv:1612.02825] [INSPIRE].
S. Schmiemann, J. Harz, B. Herrmann, M. Klasen and K. Kovařík, Squark-pair annihilation into quarks at next-to-leading order, Phys. Rev. D 99 (2019) 095015 [arXiv:1903.10998] [INSPIRE].
J. Branahl, J. Harz, B. Herrmann, M. Klasen, K. Kovařík and S. Schmiemann, SUSY-QCD corrected and Sommerfeld enhanced stau annihilation into heavy quarks with scheme and scale uncertainties, Phys. Rev. D 100 (2019) 115003 [arXiv:1909.09527] [INSPIRE].
T.R. Slatyer, The Sommerfeld enhancement for dark matter with an excited state, JCAP 02 (2010) 028 [arXiv:0910.5713] [INSPIRE].
M. Beneke, C. Hellmann and P. Ruiz-Femenia, Non-relativistic pair annihilation of nearly mass degenerate neutralinos and charginos III. Computation of the Sommerfeld enhancements, JHEP 05 (2015) 115 [arXiv:1411.6924] [INSPIRE].
K. Blum, R. Sato and T.R. Slatyer, Self-consistent calculation of the Sommerfeld enhancement, JCAP 06 (2016) 021 [arXiv:1603.01383] [INSPIRE].
E. Braaten, E. Johnson and H. Zhang, Zero-range effective field theory for resonant wino dark matter. Part I. Framework, JHEP 11 (2017) 108 [arXiv:1706.02253] [INSPIRE].
E. Braaten, E. Johnson and H. Zhang, Zero-range effective field theory for resonant wino dark matter. Part II. Coulomb resummation, JHEP 02 (2018) 150 [arXiv:1708.07155] [INSPIRE].
E. Braaten, E. Johnson and H. Zhang, Zero-range effective field theory for resonant wino dark matter. Part III. Annihilation effects, JHEP 05 (2018) 062 [arXiv:1712.07142] [INSPIRE].
Y.-L. Tang and G.-L. Zhou, Calculations of the Sommerfeld effect in a unified wave function framework, Phys. Rev. D 99 (2019) 036016 [arXiv:1806.10124] [INSPIRE].
M. Beneke, A. Broggio, C. Hasner and M. Vollmann, Energetic γ-rays from TeV scale dark matter annihilation resummed, Phys. Lett. B 786 (2018) 347 [arXiv:1805.07367] [INSPIRE].
M. Beneke, A. Broggio, C. Hasner, K. Urban and M. Vollmann, Resummed photon spectrum from dark matter annihilation for intermediate and narrow energy resolution, JHEP 08 (2019) 103 [Erratum ibid. 07 (2020) 145] [arXiv:1903.08702] [INSPIRE].
M. Beneke, C. Hasner, K. Urban and M. Vollmann, Precise yield of high-energy photons from Higgsino dark matter annihilation, JHEP 03 (2020) 030 [arXiv:1912.02034] [INSPIRE].
B. von Harling and K. Petraki, Bound-state formation for thermal relic dark matter and unitarity, JCAP 12 (2014) 033 [arXiv:1407.7874] [INSPIRE].
A. Mitridate, M. Redi, J. Smirnov and A. Strumia, Cosmological implications of dark matter bound states, JCAP 05 (2017) 006 [arXiv:1702.01141] [INSPIRE].
J. Ellis, J.L. Evans, F. Luo and K.A. Olive, Scenarios for gluino coannihilation, JHEP 02 (2016) 071 [arXiv:1510.03498] [INSPIRE].
S.P. Liew and F. Luo, Effects of QCD bound states on dark matter relic abundance, JHEP 02 (2017) 091 [arXiv:1611.08133] [INSPIRE].
H. Fukuda, F. Luo and S. Shirai, How heavy can neutralino dark matter be?, JHEP 04 (2019) 107 [arXiv:1812.02066] [INSPIRE].
J. Harz and K. Petraki, Radiative bound-state formation in unbroken perturbative non-Abelian theories and implications for dark matter, JHEP 07 (2018) 096 [arXiv:1805.01200] [INSPIRE].
J. Harz and K. Petraki, Higgs enhancement for the dark matter relic density, Phys. Rev. D 97 (2018) 075041 [arXiv:1711.03552] [INSPIRE].
J. Harz and K. Petraki, Higgs-mediated bound states in dark-matter models, JHEP 04 (2019) 130 [arXiv:1901.10030] [INSPIRE].
S. Tulin and H.-B. Yu, Dark matter self-interactions and small scale structure, Phys. Rept. 730 (2018) 1 [arXiv:1705.02358] [INSPIRE].
M.R. Buckley and P.J. Fox, Dark matter self-interactions and light force carriers, Phys. Rev. D 81 (2010) 083522 [arXiv:0911.3898] [INSPIRE].
L.G. van den Aarssen, T. Bringmann and C. Pfrommer, Is dark matter with long-range interactions a solution to all small-scale problems of ΛCDM cosmology?, Phys. Rev. Lett. 109 (2012) 231301 [arXiv:1205.5809] [INSPIRE].
S. Tulin, H.-B. Yu and K.M. Zurek, Beyond collisionless dark matter: particle physics dynamics for dark matter halo structure, Phys. Rev. D 87 (2013) 115007 [arXiv:1302.3898] [INSPIRE].
T. Bringmann, F. Kahlhoefer, K. Schmidt-Hoberg and P. Walia, Strong constraints on self-interacting dark matter with light mediators, Phys. Rev. Lett. 118 (2017) 141802 [arXiv:1612.00845] [INSPIRE].
H. An, M.B. Wise and Y. Zhang, Effects of bound states on dark matter annihilation, Phys. Rev. D 93 (2016) 115020 [arXiv:1604.01776] [INSPIRE].
I. Baldes and K. Petraki, Asymmetric thermal-relic dark matter: Sommerfeld-enhanced freeze-out, annihilation signals and unitarity bounds, JCAP 09 (2017) 028 [arXiv:1703.00478] [INSPIRE].
I. Baldes, M. Cirelli, P. Panci, K. Petraki, F. Sala and M. Taoso, Asymmetric dark matter: residual annihilations and self-interactions, SciPost Phys. 4 (2018) 041 [arXiv:1712.07489] [INSPIRE].
A. Kamada, K. Kaneta, K. Yanagi and H.-B. Yu, Self-interacting dark matter and muon g − 2 in a gauged \( \mathrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} \) model, JHEP 06 (2018) 117 [arXiv:1805.00651] [INSPIRE].
A. Kamada, M. Yamada and T.T. Yanagida, Self-interacting dark matter with a vector mediator: kinetic mixing with the \( \mathrm{U}{(1)}_{{\left(B-L\right)}_3} \) gauge boson, JHEP 03 (2019) 021 [arXiv:1811.02567] [INSPIRE].
S. Matsumoto, Y.-L.S. Tsai and P.-Y. Tseng, Light fermionic WIMP dark matter with light scalar mediator, JHEP 07 (2019) 050 [arXiv:1811.03292] [INSPIRE].
A. Kamada, M. Yamada and T.T. Yanagida, Unification for darkly charged dark matter, Phys. Rev. D 102 (2020) 015012 [arXiv:1908.00207] [INSPIRE].
P. Ko, T. Matsui and Y.-L. Tang, Dark matter bound state formation in fermionic Z2 DM model with light dark photon and dark Higgs boson, arXiv:1910.04311 [INSPIRE].
A. Kamada, M. Kaplinghat, A.B. Pace and H.-B. Yu, How the self-interacting dark matter model explains the diverse galactic rotation curves, Phys. Rev. Lett. 119 (2017) 111102 [arXiv:1611.02716] [INSPIRE].
M. Kaplinghat, T. Ren and H.-B. Yu, Dark matter cores and cusps in spiral galaxies and their explanations, JCAP 06 (2020) 027 [arXiv:1911.00544] [INSPIRE].
T. Binder, M. Gustafsson, A. Kamada, S.M.R. Sandner and M. Wiesner, Reannihilation of self-interacting dark matter, Phys. Rev. D 97 (2018) 123004 [arXiv:1712.01246] [INSPIRE].
T. Bringmann, F. Kahlhoefer, K. Schmidt-Hoberg and P. Walia, Converting nonrelativistic dark matter to radiation, Phys. Rev. D 98 (2018) 023543 [arXiv:1803.03644] [INSPIRE].
T. Binder, K. Mukaida and K. Petraki, Rapid bound-state formation of dark matter in the early universe, Phys. Rev. Lett. 124 (2020) 161102 [arXiv:1910.11288] [INSPIRE].
K. Griest and D. Seckel, Three exceptions in the calculation of relic abundances, Phys. Rev. D 43 (1991) 3191 [INSPIRE].
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].
N. Brambilla, M.A. Escobedo, J. Ghiglieri and A. Vairo, Thermal width and quarkonium dissociation by inelastic parton scattering, JHEP 05 (2013) 130 [arXiv:1303.6097] [INSPIRE].
A. Rothkopf, Heavy quarkonium in extreme conditions, Phys. Rept. 858 (2020) 1 [arXiv:1912.02253] [INSPIRE].
T. Binder, L. Covi and K. Mukaida, Dark matter Sommerfeld-enhanced annihilation and bound-state decay at finite temperature, Phys. Rev. D 98 (2018) 115023 [arXiv:1808.06472] [INSPIRE].
A. Pineda and J. Soto, Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. B Proc. Suppl. 64 (1998) 428 [hep-ph/9707481] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Potential NRQCD: an effective theory for heavy quarkonium, Nucl. Phys. B 566 (2000) 275 [hep-ph/9907240] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys. 77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
N. Brambilla, M.A. Escobedo, J. Soto and A. Vairo, Quarkonium suppression in heavy-ion collisions: an open quantum system approach, Phys. Rev. D 96 (2017) 034021 [arXiv:1612.07248] [INSPIRE].
R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jap. 12 (1957) 570 [INSPIRE].
P.C. Martin and J.S. Schwinger, Theory of many particle systems. 1, Phys. Rev. 115 (1959) 1342 [INSPIRE].
K. Petraki, M. Postma and M. Wiechers, Dark-matter bound states from Feynman diagrams, JHEP 06 (2015) 128 [arXiv:1505.00109] [INSPIRE].
P. Asadi, M. Baumgart, P.J. Fitzpatrick, E. Krupczak and T.R. Slatyer, Capture and decay of electroweak WIMPonium, JCAP 02 (2017) 005 [arXiv:1610.07617] [INSPIRE].
K. Petraki, M. Postma and J. de Vries, Radiative bound-state-formation cross-sections for dark matter interacting via a Yukawa potential, JHEP 04 (2017) 077 [arXiv:1611.01394] [INSPIRE].
E. Braaten and R.D. Pisarski, Soft amplitudes in hot gauge theories: a general analysis, Nucl. Phys. B 337 (1990) 569 [INSPIRE].
J. Pirenne, The proper field and the interaction of Dirac particles. I–III, Arch. Sci. Phys. Nat. 28 (1946) 233.
J.A. Wheeler, Polyelectrons, Ann. N.Y. Acad. Sci. 48 (1946) 219.
A. Ore and J.L. Powell, Three photon annihilation of an electron-positron pair, Phys. Rev. 75 (1949) 1696 [INSPIRE].
S.M. Bilenky, V.H. Nguyen, L.L. Nemenov and F.G. Tkebuchava, Production and decay of (muon-plus muon-minus)-atoms, Yad. Fiz. 10 (1969) 812 [INSPIRE].
T. Bringmann, J. Edsjö, P. Gondolo, P. Ullio and L. Bergström, DarkSUSY 6: an advanced tool to compute dark matter properties numerically, JCAP 07 (2018) 033 [arXiv:1802.03399] [INSPIRE].
E. Braaten, H.W. Hammer and G.P. Lepage, Open effective field theories from deeply inelastic reactions, Phys. Rev. D 94 (2016) 056006 [arXiv:1607.02939] [INSPIRE].
X. Yao and T. Mehen, Quarkonium in-medium transport equation derived from first principles, Phys. Rev. D 99 (2019) 096028 [arXiv:1811.07027] [INSPIRE].
X. Yao and B. Müller, Quarkonium inside the quark-gluon plasma: diffusion, dissociation, recombination and energy loss, Phys. Rev. D 100 (2019) 014008 [arXiv:1811.09644] [INSPIRE].
N. Brambilla, M.A. Escobedo, J. Soto and A. Vairo, Heavy quarkonium suppression in a fireball, Phys. Rev. D 97 (2018) 074009 [arXiv:1711.04515] [INSPIRE].
N. Brambilla, M.A. Escobedo, A. Vairo and P. Vander Griend, Transport coefficients from in medium quarkonium dynamics, Phys. Rev. D 100 (2019) 054025 [arXiv:1903.08063] [INSPIRE].
S. Biondini and M. Laine, Re-derived overclosure bound for the inert doublet model, JHEP 08 (2017) 047 [arXiv:1706.01894] [INSPIRE].
S. Biondini, Bound-state effects for dark matter with Higgs-like mediators, JHEP 06 (2018) 104 [arXiv:1805.00353] [INSPIRE].
S. Biondini and M. Laine, Thermal dark matter co-annihilating with a strongly interacting scalar, JHEP 04 (2018) 072 [arXiv:1801.05821] [INSPIRE].
S. Biondini and S. Vogl, Coloured coannihilations: dark matter phenomenology meets non-relativistic EFTs, JHEP 02 (2019) 016 [arXiv:1811.02581] [INSPIRE].
S. Kim and M. Laine, Studies of a thermally averaged p-wave Sommerfeld factor, Phys. Lett. B 795 (2019) 469 [arXiv:1904.07882] [INSPIRE].
S. Biondini and S. Vogl, Scalar dark matter coannihilating with a coloured fermion, JHEP 11 (2019) 147 [arXiv:1907.05766] [INSPIRE].
M. Laine, O. Philipsen, P. Romatschke and M. Tassler, Real-time static potential in hot QCD, JHEP 03 (2007) 054 [hep-ph/0611300] [INSPIRE].
N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky, Static quark-antiquark pairs at finite temperature, Phys. Rev. D 78 (2008) 014017 [arXiv:0804.0993] [INSPIRE].
S. Kim and M. Laine, On thermal corrections to near-threshold annihilation, JCAP 01 (2017) 013 [arXiv:1609.00474] [INSPIRE].
S. Kim and M. Laine, Rapid thermal co-annihilation through bound states in QCD, JHEP 07 (2016) 143 [arXiv:1602.08105] [INSPIRE].
CMS collaboration, Measurement of nuclear modification factors of ϒ(1S), ϒ(2S), and ϒ(3S) mesons in Pb-Pb collisions at \( \sqrt{s_{NN}} \) = 5.02 TeV, Phys. Lett. B 790 (2019) 270 [arXiv:1805.09215] [INSPIRE].
S. Biondini, S. Kim and M. Laine, Non-relativistic susceptibility and a dark matter application, JCAP 10 (2019) 078 [arXiv:1908.07541] [INSPIRE].
J. Smirnov and J.F. Beacom, TeV-scale thermal WIMPs: unitarity and its consequences, Phys. Rev. D 100 (2019) 043029 [arXiv:1904.11503] [INSPIRE].
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Binder, T., Blobel, B., Harz, J. et al. Dark matter bound-state formation at higher order: a non-equilibrium quantum field theory approach. J. High Energ. Phys. 2020, 86 (2020). https://doi.org/10.1007/JHEP09(2020)086
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DOI: https://doi.org/10.1007/JHEP09(2020)086