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DOI: 10.1007/JHEP09(2015)127

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Citation for published version (APA): Ellis, J., Luo, F., & Olive, K. A. (2015). Gluino coannihilation revisited. Journal of High Energy Physics, 2015(9), [127]. https://doi.org/10.1007/JHEP09(2015)127

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Download date: 30. Sep. 2021 JHEP09(2015)127 , χ . We ˜ q m Springer April 13, 2015 August 9, 2015 : August 10, 2015 : : September 21, 2015 : Revised Received 10.1007/JHEP09(2015)127 Accepted 8 TeV for a wide range of Published ∼ doi: for which the relic cold dark χ m Published for SISSA by [email protected] , c,d [email protected] , and Keith A. Olive 1, and that the LSP may weigh up to b . . 3 is identified as the lightest supersymmetric particle (LSP), the gluino = 1 χ ˜ g 1503.07142 The Authors. /m ˜ q Feng Luo c Supersymmetry Phenomenology

Some variants of the MSSM feature a strip in parameter space where the m , a,b [email protected] 100. . ˜ g 50% if /m ∼ ˜ q is the next-to-lightest supersymmetric particle (NLSP) and is nearly degenerate with E-mail: School of Physics andMinneapolis, Astronomy, MN University 55455, of U.S.A. Minnesota, William I. Fine TheoreticalUniversity Physics of Institute, Minnesota, School ofMinneapolis, Physics MN and 55455, Astronomy, U.S.A. Theoretical Particle Physics and CosmologyKing’s Group, College Department London, of London Physics, WC2RTheory 2LS, Division, United CERN, Kingdom CH-1211 Geneva 23, Switzerland b c d a ˜ Open Access Article funded by SCOAP Keywords: ArXiv ePrint: find that bound-state effectsmatter can density increase lies the within maximum as the range favoured bym astrophysics and cosmology by as much and the relic cold dark mattercosmology density by is brought coannihilation into the withthis range the allowed gluino by gluino coannihilation astrophysics and strip NLSP. instates We calculate the and MSSM, initial-state the including Sommerfeld relic theof enhancement, density effects the and along of taking gluino gluino-gluino into and bound account LSP the densities decoupling that occurs for large values of the squark mass Abstract: lightest neutralino g John Ellis, Gluino coannihilation revisited JHEP09(2015)127 23 ], it is 2 , 1 22 parity is violated, in which case R 4 – 1 – ] that the conventional missing-energy searches at 4 , 3 3 9 parity is conserved, but supersymmetric particles are too heavy 12 R 8 20 4 19 7 13 1 21 There are two issues with this last possibility. One is the accentuation of the problem 6.3 Higgsino LSP 6.1 Bino LSP 6.2 Wino LSP 3.1 Dissociation 3.2 Formation was postulated to mitigate,The and cold the dark other mattersupersymmetric is particle may (LSP). the well However, even cosmological not ifonly cold the consist as cold dark only, dark an matter matter or upper densitymass density. even is limit that considered predominantly, on depends of the on the the relic specific lightest LSP LSP density, candidate it under imposes consideration. an upper bound on the LSP the LHC have beensupersymmetry unable may be to hiding resolve?processes? among Or the Or perhaps jets perhaps to and have leptons been produced detected by during Standard Run Model 1 of theof LHC? the naturalness (or fine-tuning) of the electroweak scale that low-scale supersymmetry 1 Introduction In the absencenatural of to any ask how signal andsight for where with supersymmetry supersymmetry a may during compressed be hiding. spectrum Run [ Perhaps 1 it of is hiding the in LHC plain [ B List of the transition amplitudes for the gluino (co)annihilation processes 7 Summary A The procedure for obtaining a thermally averaged cross section 5 Boltzmann equations 6 Numerical results 3 Gluino-gluino bound-state formation 4 Conversion rates Contents 1 Introduction 2 Sommerfeld enhancement JHEP09(2015)127 – ]. ]. -, 8 7 s 30 20% ∼ ]. Here we 6 , 5 100, the densities 1, falling to . ∼ > ˜ g = 1 could be, if it is nearly ˜ g /m χ ˜ q /m m ˜ q m and the gravitino [ 1, due to cancellations between . This is of relevance to assessing, χ g → ˜ g the possible LSP mass. There is also a 50% for ∼ /m ]. In general, one would expect that the ˜ q 29 m – LSP masses than would have been estimated in – 2 – reducing 19 , 17 larger -conserving supersymmetry. There can of course be no cast- R ], and the main new elements here are in our discussions of the 29 – 19 , 17 10 to 50. Another effect we discuss is that, if ∼ ˜ g parity is conserved and the LSP is present in the Universe today as a relic from -channel diagrams that tend to reduce annihilation rates. u R /m ˜ q As already mentioned, there have been several previous analyses of neutralino-gluino We study here the question how heavy the neutralino LSP The relic LSP density depends not only on the LSP mass, but also on the rates at which If m ], and also the case of the gluino [ - and for of neutralinos and gluinossignificantly decouple higher relic and density, coannihilation thereby reduction effects in freeze-out the possible early, LSP leavingt mass a for small into the LSP as isestimates usually of assumed the in bound-state discussions productionrelic of rate, dark coannihilation. and matter find We density that, presentstate is numerical for formation substantially fixed is reduced sparticle neglected. masses, comparedcosmological the Conversely, with range the the for cosmological cases substantially relic where densitythe bound- may absence lie of within bound-state the effects: this effect is tained in the presenceattention of to a the largemodels coannihilation squark (with to processes for gluino and example, massAs radiative leave electroweak we ratio. their symmetry discuss breaking) in Here, application for detail,that we to future bound may will work more states subsequently restrict [ can complete annihilate our remove from into the Standard primordial Model plasma particles, gluino pairs before they can decay neutralino LSP and thescope gluino. of However, this a paper. complete analysis of this casecoannihilation lies [ beyond the effects of gluino-gluino bound states and of the issue whether coannihilations can be main- for example, what centre-of-massbe energy ‘guaranteed’ would to be detect needediron for guarantee, a even proton-proton within collidercase the to studied MSSM. here theLSP For neutralino example, mass LSP limit even mass in could limit the be depends gluino substantially on modified coannihilation the if squark the masses, squarks and were the degenerate with the 18 heaviest LSP will be allowedcharge, when it namely coannihilates the with gluino. the particle with the largest colour degenerate with, and coannihilates with, the gluino ˜ it annihilated with itselfOther and things coannihilated being with equal, otherare the sparticles maximised, largest in which LSP the happens mass early whenthere the is Universe is LSP [ allowed only is when one (nearly) such such degenerate coannihilating coannihilationbe with sparticle other rates species, maximised particles. the for coannihilation If rate a willcases coloured in where general sparticle. the coannihilating There particle have is been a analyses squark, specifically in the the lighter literature stop of squark the [ the Big Bang, itactions. is expected In to the bemost minimal electromagnetically plausible supersymmetric neutral candidates extension and areconsider of have the the only the lightest neutralino weak Standard neutralino case, inter- Model and the (MSSM), cosmological the upper bound on its mass. JHEP09(2015)127 (2.1) (2.2) (2.3) , we discuss the representations, summarises our 8 TeV if it is the 3 s 7 ∼ , , we review the Sommerfeld en- , β α  2 0 i contains some numerical results for C ..., ≡ 100. If the LSP is the neutral Wino, 6 − + 2 cross sections needed for solving the s i 1 ∼ < : C − → are the quadratic Casimir coefficients of the ˜ g 0 i − bx πs C e f is the quadratic Casimir coefficient of a specific /m ] πs + ˜ q – 3 – C f − − a  38 m C 1 – s and r i 2 = α ∼ < 36 is the velocity of one of the annihilating particles in ≡ C i β ) = s σv ( = 8. 7 TeV, and for a neutral Higgsino the upper limit it be- h V F f ∼ C 0, and α < ], Sommerfeld effects such as these have been implemented in the , we consider the rates for conversion between gluinos and neutrali- = 3, or 16 4 f C , we present and discuss the coupled Boltzmann equations for neutralinos = 3. The relevant final states are in singlet, octet, or 27 5 = 0, for calculating the relic dark matter density. In the coannihilation region of 8 f C 1 ] C is the strong coupling strength, 6 TeV. is the coefficient of a Coulomb-like potential whose sign convention is such that = 35 s 0 – i code ∼ α α C 31 As discussed in [ The layout of this paper is as follows. In section Taking these effects into account, we find a maximum LSP mass Information about this code is available from K. A. Olive: it contains important contributions from = 1 i , gluinos and gluino-gluino bound states. Section interest, this code uses a non-relativistic expansion for annihilation cross-sections: T. Falk, A. Ferstl, G.Spanos, Ganis, and F. M. Luo, Srednicki. A. Mustafayev, J. McDonald, K. A. Olive, P. Sandick, Y. Santoso, V. annihilating coloured particles. InC the case of octetfor annihilating which particles such as gluinos, SSARD where final-state colour representation, and where the attractive case has the centre-of-mass frame of theCoulomb-like collision. potential has In the the form cases [ of strongly-interacting particles, the Before discussing the formationbriefly and Sommerfeld effects of effects gluino-gluino inrates bound gluino-gluino at states, annihilation, low we velocities, which first andgluino. may discuss are enhance particularly As annihilation relevant in ators the general [ case rule, of the initial-state strongly-interacting interactions modify s-wave cross-sections by fac- technical aspects of theBoltzmann computation equations. of the 2 2 Sommerfeld enhancement formation of gluino-gluino bound states,Universe. considering also In dissociation section processes innos. the early In section χ the gluino coannihilation strip andconclusions and a discusses discussion their of significance its for endpoint. future colliders. Section Finally, appendices present Bino, which may bethe attained upper for limit 10 iscomes reduced to hancement in the relevant gluino-gluino annihilation processes. In section JHEP09(2015)127 /r s and α (3.1) (3.2) 3 gg − following → ˜ g dis ) = g r σ ( V = 0 and spin angular . In principle, one may ) so that it acquires a ]. Gluino-gluino bound ˜ L R 2.3 28 m − . A Sommerfeld enhancement ˜ g /x m , . The normalised spatial part of 2 ˜ g ≡ r/a m 2 − . B e ˜ g 2) bound state decays predominantly to a E 2 / m / s s 5 s 1 . α − α
T (3 3 4 243 . These processes become important when the . – 4 – ' πa g B = B + ˜ R E ˜ ) = R Γ r ( ↔ bs φ g state, namely , with orbital angular momentum s + ˜ ˜ bound state is R g ), modifying the leading term in ( s 2.1 represent the contributions of the s- and p-wave cross-sections, . The expressions for the matrix elements for the coannihilation b is the gluino-neutralino mass difference [ ) is the Bohr radius. The 1 A ˜ g 0 in ( m m and s . In this paper we have included these enhancements in the ˜ , we first calculate the bound-state dissociation cross section, ], where the photoelectric effect for an atom is calculated. The central part x α a c = 0, which is expected to be the most copiously produced. Since we are g √ α < 39 (3 / S + denotes an average over the thermal distributions of the annihilating particles, ∝ 2 state, with binding energy ˜ i R cross sections. The procedure for obtaining a thermally averaged cross section , and the dots represent terms of higher order in 1 s ≡ ¯ q ... ↔ a h q b = 0 where ∆ g m/T The dominant process for the formation and dissociation of gluino-gluino bound states → in the thermal plasma is ˜ + ˜ ≡ m ˜ a g ˜ ˜ ˜ 3.1 Dissociation In order to calculateg bound-state formation andsection dissociation 56 via of [ the dominant processes where pair of gluons and the leading order decay rate is considering gluinos weighing several TeV, wetheory expect the to leading be order of a QCDfor useful perturbation the approximation, 1 and assumethe the wave function Coulomb for potential this 1 to the binding energy ofform the colour-octet states asbound well as with singlets, larger but wavelightest the functions colour-singlet latter at state, are the expected 1 origin.momentum to be Here more we deeply focus on the production of the relic density by aand factor much larger that than is the non-negligible uncertainty in relative the to cosmologicalR the cold Sommerfeld dark matter enhancement, density. plasma temperature falls low enough for typical thermal energies to become comparable thereby lower the final neutralino relicabove abundance. further increases The the Sommerfeld cross enhancement section discussed of in specific neutralinos channels and allowing again for lowers the∆ larger abundance masses at thestates tip can of further the serve coannihilation to strip remove defined gluinos by from the thermal bath and thereby lower the is given in appendix processes are given in detail in appendix 3 Gluino-gluino bound-state formation Gluino-neutralino coannihilations may increase the effective annihilation cross section and the coefficient x occurs when singularity g where JHEP09(2015)127 , - ]: C /r ) = s b 39 (3.8) (3.5) (3.6) (3.7) (3.3) (3.4) α ˜ g 3 2 a is real) are the g − (˜ ) can be L k ~ . The δ C ( ], while the + is the same 3.4 − ) = | L 1.) 40 r and ~p , the result is iδ is ( | − c f − b = 0 ~ /r e V ˜ g = s , a α 1 g P 3 PC − J = 0. (Note also that − . κ 1, and more generally . ) ), and S r  , and because the gluon ( ~r /κξ r = 0 | ) = bs · πξ , which needs to be in an ~p r 2 s f φ | ~p ~r, ( − α 3 φ , , e  V r > a = 0 has ≡ d ~p i B . i L arctan(1 − φ Ω S E φ ξ = 1 and P ~r d · 1  4 ) 2 k ~ r + ~r i r − | L | ( · e e 2 ˜ fi ~p g ] | | = 0 and ~p pL 2 ! ~p ]. ) m | )  R 2 |M c rel 2) 39 ξ 1, which is justified because the bound- 1 L | ~ v κξ = 0 and / = pair and ~p , with iδ P ˜ g | ) b ) ) = 1 and the gluon has A − ≈ L r ~ ˜ g m ∇ · e f 2) B -parity) conservation requires that ( ( (1 + ~r / πω φ 1 · a i E C – 5 – 1 for ˜ [1 + ( g ( state, the Coulomb potential is p 2 is 8 k ~ g i − 4 P R − m 2 e A + 1) f satisfies the conservation condition | ( 

 ˜ g ω φ ~p state is the same as that of the gluon [ , and the second term on the l.h.s. of ( s L 3 | ) ∗ f ω B s m α 2 ω A rel φ (2 2 α E v L = (3 i =  Z , we may discard the phase shift factor 2 (2 2) state is opposite of that of the gluon, for all color indices. 2 2 f / | =0 0 ], namely = dis ∞ = a ˜ g s φ + X L fi s | a 39 m dσ fi ω α ~p B 1 | 2 |M 2 E ground state with π M ) is exponentially suppressed for 3 9 r s = 6, so that the dipole approximation should be sufficient for our 2 ' ( . f 0 bs = φ φ ωa < , where its energy 0 dis ω σ . Therefore we write rel L = v i | is the wave function for an 8 k ~ = 1 term survives, due to the selection rule from the dipole approximation. | is a wave function for the Coulomb potential is calculated following section 56 of [ f i ), where the 1 φ is the momentum of one of the annihilating gluinos. (Note that L is the wave function of the free ˜ fi c φ | g f ~p ( | 1 when φ M C ) Since The normalised spatial part of the wave function for the free pair ˜ The dipole approximation imposes a selection rule on We use the dipole approximation, ˜ R = 1 state. Further, charge conjugation ( ( -parity of colour-symmetric 8 whereas different from eq. (56.12) of [ so that where Only the Since we wish to calculate and the factor Therefore, the only possibleparity state is conserved for in this case, because L C parity of the colour anti-symmetric 8 C where as the relative momentum,neglected.) ( We find ωa < purposes. state wave function momentum where polarisation and momentum vectors of the gluon, respectively. of the calculation is the evaluation of the transition amplitude given in eq. (56.2) of [ JHEP09(2015)127 = 0 (3.9) . As (3.11) (3.12) (3.13) (3.14) (3.10) S 8, and dis ) by the σ √ / 3.3 de = 4 into the δ 2 . | . are products of = 2 ˜  R √ r/a ~r r 2 | − · ~p e color √ | ~p | 2 i, /  φ 1 = 1 state, and the wave 1 − ) is calculated from the ] ) P 3 L ) . 3.3 r 42 ( , πa 1 ( p = 3 41 × R 2 |

2 ~p , 3 | ch b √ 2 δ ˜ g ). 3 f . µ , µ × cd 3 γ 2 1 δ )] = in the transition amplitude ( a 2 √ ¯ fγ ¯ ˜ g √ π √ 8 f µ / pair and the bound state 1 = 3 ) is compensated by a factor of 2 due to the ) + abc A Q √ b = f

f ˜ g = 0 state, and the wave function needs to be  c µ abd ) – 6 – a g 3.10 = f θ, φ eQ s g π L 2 g −

− abc i + 2 ↑↓ − ↓↑ f ( = cda 2 in ( = f / r, π L θ, φ ( ]: L 3 bs had − = 1. 41 √ φ f κ 8 1 r, π ) + √ (

f = 2. This equation is averaged over the gluon polarizations and φ factor in the transition amplitude ( κ − ~r · r, θ, φ ) ( k ~ i bs e and ) φ [ 2) , since the factor 1 rel c 3 (the latter because 2 / r, θ, φ ~ 1 ˜ g ( √ /v √ abc ~ ∇· f m / s f ( φ α i  3 2 hae 2 − f 1 √ = = ξ Concerning the spatial part of the wave function, we need to take into account the The ( The total wave functions for the free ˜ color f, The coefficients in thesemodulus-squared two of equations the introduce spatial an wave function extra factors. factor Finally, of recall that we have averaged On the other hand,function the needs final to free be pair antisymmetrised: is in the antisymmetric Note that all color indices are summed over. fact that both the initialbound and state, final they states contain aresymmetrised two in as identical particles. in the eq. symmetric In (2.14) the of case [ of the colour factor Majorana nature of the gluino. Putting the above colour factors together, we obtain which can be compared with the corresponding QED interaction Lagrangian We simply replace the electric charge factor φ gluino-gluino-gluon interaction Lagrangian state, the spin wave functions are both and the spin partsfor of the the colour wave part functions of do not the introduce wave any function, extra according factor to in [ would reduce to eq. (56.12) if the spin, colour and spatialthe parts gluinos, of the the wave total functions.of wave functions In the view should wave of function, be the since anti-symmetric. Majorana both Concerning nature the of the bound spin state part and the free gluino pair are in an where JHEP09(2015)127 dis σ (3.17) (3.19) (3.20) (3.21) (3.15) (3.16) (3.18) , is given in π d g 27 c 512 g 2 comes from the , ) s → 1 ˜ 2 g eq g b − m πα ˜ g 1 dn a (4 ω/T ). Using g e dis ) , which is related to σ ! T ˜ Hγ ) 2 eq R ξ
/ bsf n 2 3 ξ σ 2 rel ↔ ( 2 3 v π = p . 2 − , , 2)( e . − / rel e 44 ˜ g 0 dis π . − 1 dis 16 2 dv 1 σ m σ ( 1 ) 2 − × 1 2 ≈ ] = rel e , v 1 2 2 rel ( + 2 2 rel e 8) v 1 2 32 f 3 ω × g × 2) πξ πv 8) −  2 g 1 8 / 4 dω = 0 limit we find ˜ 1 ˜ g 2 − R × (2 2 ) πξ e / bound states is a non-negligible component in g × ω m – 7 – ω/T − 3 2 2 → × (2 4 rel e ˜ g − [( 1  v 1 2 ˜ g g 3 1 rel 2 × rel ) g ω/T v / π v 1 3 = π e ann ˜ g 4 πT = σ g 2 (2 bsf + = 3 m ( g 2 ˜ g g σ ) ˜ g 1 + g  R bsf ξ dis ann g σ  (2 ) σ S = π ξ 2 rel ) = v (2 eq g − ]: e rel π v 28 2 dn bsf ( − σ f 1 ˜ eq g 6 1 n

˜ . Therefore, in the eq g n rel 1 2 ) = /v s on the l.h.s. of the above equation is introduced to avoid double-counting 8. Therefore, the final dissociation cross section is rel / α v 1 2 3 2 ann = ), we find σ ( ξ 3.4 ann S Therefore, we see thatdetermining the the inclusion final of neutralino relic ˜ density. eqs. (2.13) and (2.25) of [ where where For comparison, the Sommerfeld enhanced s-wave cross section for ˜ and ( enhancement of bound-state formation due tobackground the (similar stimulated to gluon the emission in stimulated the recombination thermal in where the the number of bound-state formation reactions, and the factor space integration. 3.2 Formation We come finally tothrough the the bound-state Milne formation relation: cross section, a factor of 1 where the final factor of 1/2 is to avoid double counting of gluinos in the final-state phase- over the polarisations of the gluon, but we also need to average over its colour. This gives JHEP09(2015)127 , , t 2 6 ) for 2 , are p 1 2 2 , + 1 . Due to ˜ , while 1 χ H 2 p ) 2 p > m = ( ˜ g s − m 1 p = ( , and the gluino decays s ¯ q ˜ g and its charge-conjugated ¯ ↔ u d ¯ q , as well as the corresponding + χ ¯ u , except that the expressions in χ , d + are identical to the corresponding ˜ B gq → χ 2 g stands here for all six quark flavors, ↔ ]. However, as we show below in section , ˜ ↔ q ¯ q |T | 45 χq g – χq 43 → and ˜ g ¯ d 100) [ – 8 – ˜ g are so close in mass that all the gluino three-body decays ) is approximately equal to the equilibrium ratio  ↔ χ 100, the gluino-neutralino conversion rate dominates over the for the coannihilation processes, whereas it is ¯ LSP u + 1 4 . and , whereas for the gluino decays, /n χ 2 g ) , = 4 . When the neutralino is a Wino or Higgsino, the processes ¯ p q ˜ gu 1 2 NLSP n − χq × ↔ 1 1 2 p d ↔ + g χ = ( u , are also included. We note that − and χ 2 stand for all the three generations of up-type and down-type quarks. Also, . ) should be multiplied by a factor of 2, because the statistical factor for the 2 3 p The squared transition matrix elements B u, d |T | is a Higgsino, the two lightest mass-degenerate neutralino components, 2 − do not change. 1 χ p u The gluino decay rates are then obtained by performing the standard 3-body phase We calculate the gluino decay rates for ˜ The interconversion processes we consider are Since we treat quarks as massless, three-body decays will always occur as long as 2 = ( logarithmic corrections, the gluino two-bodythe decay squark-to-gluino into mass a neutralino ratiogluino and is coannihilation a becomes very gluon irrelevant large becomeswhen for important ( the such when a squark-to-gluino large massgluino squark-to-gluino ratio two-body mass is decay ratio. ratewould On when be the the kinematically other forbidden, ˜ hand, if quarks are not treated as massless. space integration. Thebecause inverse-decay they processes are do takenthe not into next have account section. to automatically be by the calculated Boltzmann separately, equations given in initial spin averaging is the gluino decay processes.should We also note be also that changed correspondingly thet as definitions follows: of the for Mandelstamand the variables coannihilations, element process. ones for the coannihilationappendix processes given in appendix involving a chargino, processes for and the when both taken into account. For each relevant process, we first calculate the transition matrix 100 as we showevolve below. separately, and coannihilation For effects are largerdifference. essentially squark shut off masses, independent of the the gluino mass and neutralino abundances and the inverse decays ˜ rates are mediated bythat light the Standard ratio Model ofand particles densities allows for and ( a are simplification inneutralinos always the and fast. Boltzmann gluinos equations. must This However, theare proceed implies interconversion via of heavy. squarks, leading Theand to requires relevance a the of suppression ratio if the of the coannihilation squark squarks masses process to relies the gluino on mass fast to conversion be rates, less than approximately For coannihilation to begluinos) effective, the must coannihilating be species inNLSP (in thermal must this contact. be case neutralinos That fastercoannihilation, and is, than connectivity the the of the rates Hubble two for rate. species interconverting can In the be both LSP taken the and for granted, familiar as cases the of conversion stop and stau 4 Conversion rates JHEP09(2015)127 ), = χ t (5.1) (5.2) (4.1) ]. m , ]. In a . The 2 q (2 ) 46 48 apply if , / 2 is pulled , ] p /E 2 q 47 u | # A q m − u ~p   | 1 − eq and χ p eq ˜ g or χ = 2 χ Y Y t Y 3 q A and their charge- m eq v q = ( χ eq ¯ ˜ d g − Y , ˜ g s Y 2 | ) − 0 q − → q for the coannihilations, ˜ g ~p ˜ + 1 g | Y ¯ u m 1 8 Y d χ should be multiplied by a 2 + /T |  Y + q -hadron was studied in [ χ q  B ˜ g s 1 E ~p , R | ˜ ˜ e g g . m i is the velocity of the incoming χ ˜ gu π i ˜ Γ R 3 s q 4 h ) n v · → σv = [( π + 2 d ≡ · (2 − h ! + min ˜ R 3 q χ
˜ g q eq ˜ g E , v Y eq ,Y χ Y c is a function of the incoming quark energy is ˜ ˜ g gq Y σ s c eq n χ q eq σ χ Y → ∞ E min Y ≡ q + – 9 – − E ˜ g − χq Z χ χ Y ,Y Y =

χ χ s q is the squared center-of-mass energy, and is given by n Y s 1 u i dn ≡ c q χχ v Γ χ or i c h , where the upper signs in the definition of Y σ t 2 σv q ) 4 X Z p − h − = ∓ " ) count the quark color and spin degrees of freedom, respectively. i ) c 1 are identical to the corresponding ones for the coannihilation processes χ p Γ , except that the expressions in appendix 4.1 h 2 m , where the lower limit of xs ( B = ( q over the Fermi-Dirac distribution of the quark in the initial state, |T | for the conversions. Also, compared to the coannihilation processes, the H E u q χ 1 3 v = c m σ and , because the factor for the initial color averaging is χ is brought into the initial state, while the lower signs apply if 8 3 represents the quark in the final state. Here is the conversion cross section for any of the relevant processes discussed above. In + 2 2 dx ) dY 0 c 2 q B 3 q ¯ σ d p m To set up a coupled set of Boltzmann equations, it is convenient to rescale the number For each of the quark flavors, the thermally-averaged conversion rate is obtained by To calculate the conversion rates for ± + A set of coupled Boltzmann equations for the photino and a gluino or . In this reference frame, 3 1 2 χ q B p ¯ different context, Boltzmann equations involving a bound state can be found, for example, in [ These are governed by the following set of coupled Boltzmann equations: We are now in a position(or set to of put equations) all in of orderthree the to separate solve components for density discussed the components: above relic into density. neutralinos, a To gluinos rate do and equation so, we bound begindensities states. by of considering neutralinos, gluinos and bound states by the entropy density, Again, the inverse conversion rates areequations. taken into account automatically by the Boltzmann 5 Boltzmann equations E m where quark, and it is relatedfactors to 3 the and energy 2 and 3-momentum in of ( the quark by where the initial neutralino or chargino rest frame, Mandelstam variables for the( conversion processes areq re-defined as over to the initial state. integrating conjugated processes, we firstmatrix calculate elements the crossgiven sections. in Again, appendix thefactor squared of transition whereas it is JHEP09(2015)127 . : are    (5.8) (5.6) (5.7) (5.3) (5.4) (5.5) ˜ g ! ˜ eq R i ˜ R Y ˜ Y Γ eq R ˜ R h Y , namely, Y 

/s   ˜ 2 R and , i eq ! χ eq ˜ g 2 χ χ i Γ Y c Y h ˜ g Y eq is the decay rate m ˜ g Γ  Y eq Y 1 2 h eq ˜ χ R ˜ g eq ˜ q i Y R Y 

Γ Y ∗ is the relative velocity h − P − − g , 1) enhancement factor − ˜ g ˜ g 1 N 2 χχ Y ˜ Y R i −   G χ 45  Y  3  Y ˜ g σv eq . Hence, to a good approxi-  ˜ g π ˜ ˜ g s  1 R h Y 2 ˜ Y g , 4 ω/T Y Y dis / ˜ g ˜ g e i Y 1 i , χ .    ( i − N Γ Γ ) ˜ gg / ˜ g ˜ h !# eq R = G dis T n annihilation into Standard Model → σv i ( !# ˜ + ˜ − h 2 eq R ˜ R /n R Γ = ˜ g x ˜ ˜ g Y bsf 2 H eq h R ˜ Y i R ) ! i ) − h χχ n Y P Y has the same effect as T eq + ˜ ˜ eq g  g eq ˜ g ˜ gg σv ( are summed over, ˜ g σv M ˜ g n Y eq ˜ g Y h ˜ eq g Y ( → ˜ g H n 1 2 χ Y ˜ Y eq R − h Y ˜ g eq ˜ g ˜ gg χ ˜ gg bsf eq ≡ Y i ˜ eq +  g i Y ˜ g → → ) Y ˜ Y eq ! ˜ ˜ − R χ R R σv − – 10 – σv ˜ g ˜ g h ˜ g − Y ˜ h , where i eq R − i ˜ m χ R Y ) is of order -10, whereas each of the terms on the χ ˜ ( g 1 2 Y ˜ + g Y ˜ g Y − Y σv σv Y Y ˜ ˜ h g

˜ R h 5.6 = ˜ R g /m − i Y ,H

Y s Y 1 P + 3 Γ are to be understood similarly, 1 

 dis h i are thermally averaged, and the superscript ‘eq’ denotes ˜ T i R c

M ˜ ˜ g g bsf s 1 s i i 5 s ˜ ˜ g g i ˜ Γ g Γ ∗ ˜ bsf i i h Γ h R α g n i h i σv ... 2 ˜ gg σv Γ σv h q h π σv h = 45 X 1 2 . Finally, → 2 ˜ R − h − h −h + + ˜ eq ) R ˜ ˜ g R = i " " Y 3.2 T and Y ) ) is the bound-state formation cross section times the relative velocity ( χ χ σv , s ˜ g h H χ χ m m i bsf xs xs T ( ( + i m ) are of order ˜ σv H H R are the total numbers of effectively massless degrees of freedom associated h σv i ≡ h ∗ Γ 5.6 = = g x h ) can be written in a more intuitive form: ˜ g ˜ R − 5.4 dx dx dY , and and = dY ˜ R s ˜ R ∗ x g Eq. ( Y ln ln d where d where One can check that ther.h.s. l.h.s. of of eq. eq. ( ( mation, we can set the two terms on the r.h.s. equal to each other and solve for of the two incomingas discussed gluinos, in taking section it into converts account the the boundthe 1 quantities states bracketed to by gluonsequilibrium without yields. altering the density of free gluinos. All with the entropy density and thetimes energy density, the respectively, totalparticles, cross section and forthe the total channels conversion for ratepossible and quark gluino and decay anti-quark rate channelsof discussed for the in the the previous section, and all and JHEP09(2015)127 s ∗ g 20. . In . (5.9) ∼ < eq (5.11) (5.12) (5.13) (5.10) χ # 20. ˜ g  /Y ∼ < eq ˜ g eq /m ˜ g ˜ g ˜ q Y Y eq m ˜ g /m 10, due to the ≈ ˜ q Y χ ) throughout the , m & − T
/Y ˜ g ( x 2 ˜ g Y ) Y H ˜ g is nearly temperature eq Y for ˜ R Y  i ( ˜ R Γ i dis h − i Γ 2 h . Γ ˜ g h i Y n  . rel + i ˜ gg is the ‘standard’ term in the first eq v j ˜ ˜ eq g g . rel ij → ˜ g Y n n ˜ v ˜ g ˜ g σ R i eq h ˜ g ˜ n gg i + by a coefficient not too much different eff i ,j Y → σ σv + eq χ ˜ h h R eq σv − ), which is true only when ˜ χ g n bsf h n 2 eq  i i j n eq ,i n s χ = + Y ∗ i σv eq σv = ˜ ,i h h dx R Y /Y which takes into account the evolution of n i dg i is sufficiently larger than  eq is related to the bound-state formation process n eq – 11 – + s Γ s ˜ ˜ g g n ∗ ∗ h ij ij i ˜ x Y R g dx i X i ˜ gg dg Γ i i 3 is much smaller than X h s Γ X σv ∗ h ˜ is interpreted as the total number density, g x h − g → ≡ i ≡ n 3 ˜ g ≡ ) = ( 1 n ˜ n and bsf rel χ R χ, ˜ gg  i v eq g i = → X ) c n /Y ˜ eff σv R ) changes substantially, which is the case when the squark i,j ˜ χ g Γ ˜ g σ ˜ g h Y ), the expression for i h m is the total equilibrium number density, − Y q −h xs ( , where " σv 5.9 eq + P H h ) n χ χ should be related to − n/s Y m xs ˜ gg ( = = → H ˜ Y R dx ˜ ) can be recast in the familiar form suitable for coannihilation calculations, g dY , where i = ), 5.9 /s ) ) can be solved by using the very good approximation σv ˜ g decreases with the decrease of temperature while h eq 5.10 Y ˜ ) combined with the second line involving the bound states. We re-emphasise g 5.9 n n + = dx 5.9 χ Y eq ( Y In eq. ( If at least one of the d line of ( that this simplification requiressection, so a that fast we interconversionFor can larger rate set squark as ( masses, discussed werelic use in density. the the coupled previous set of Boltzmann equations to solve for the The effective annihilation cross section is As one can see from eq. ( and where we have includedwith the temperature. term As we will see, this approximation is valid so long as large, eq. ( this case, eq. ( and we can write by crossing, from order 1. period during which ( mass appearing in the denominators of the matrix elements for these processes is not too Moreover, we note that fact that independent. Since the process ˜ Therefore, we find that JHEP09(2015)127 . : β ’s. χ i is a χ i (5.16) (5.15) (5.14) χ ]. , where the . In future χ ˜ q 30 m /n i χ n = , = 2, 6 and 8 for Bino, eq χ ) . ) /n eff i χ /T is the Bino. eq χ 1 g /T n χ χ 1 χ m We assume that the neutralino , ≡ i j i m χ i χ 4 i χ . r χ χ r i ∆ r β , , 0 χ ∆ 0 − i r ¯ ), ˜ q , neutral or charged, as gq χ j i − q r χ i → χ i ˜ g χ q and the squark masses, i χ i is the lightest component (i.e., the LSP) exp( 5.14 χ i → χ , , 2 ˜ i g 1 exp( ˜ g i / i i c i 2 χ 3 σv m χ eq χ , and tan / ) σv Γ Γ h 3 i 0 n n h h h j ) χ i A i i i i i ,χ and for the most part, our results do not depend – 12 – i χ χ χ χ χ χ , X ˜ X X χ X X X q µ explicitly as m i ≡ ≡ ≡ ≡ ≡ components, χ is a neutralino, and they are different when (1 + ∆ i ≡ ˜ ˜ g g r χ i c i eq χ i χ χ (1 + ∆ χχ n χ i Γ i n χ Γ g h i h eff χ σv i σv χ g h h χ g X components have the same mass, when = indicate all the possible quark and anti-quark channels for the ≡ 1), and we assume i 0 , the gluino mass, q χ χ q eff χ − r χ 1 m g χ and /m q i χ m ( ’s. We can then write i χ ≡ i is the same as χ 0 q We begin with the case in which the lightest neutralino When the LSP is a Wino or a Higgsino, we can still use all the above equations to The exception is the case of Higgsino-gluino coannihilations for which the vertices do depend on tan 4 the neutralino mass, work, we apply these resultsuniversality) to and more pure realistic gravity mediation CMSSM-like models models (without with gaugino vector-like multiplets mass [ We now present some numericalin results this obtained section using areWe the based assume above degenerate on formalism. squark simplified masses, supersymmetric Ouron spectra results supersymmetric defined parameters such atis as the a weak pure scale. state of either a Bino, Wino, or Higgsino. Thus our free parameters are simply In the limit thatWino all and the Higgsino, respectively. 6 Numerical results where ∆ among the chargino, and the conversion rates and gluino decaylatter rates. ‘=’ is In guaranteed eq. byFor ( the later fast discussion, conversion and/or it decay is rates useful among to the define different an effective number of degrees of freedom for where solve for the relic density.the All contributions we from need each to of do the is re-define the following quantities to include JHEP09(2015)127 t χ = = )– 13 14 12 ˜ g - - - m χ as a 120, 10 10 10 5.2 , . ˜ g m = 10. /m ˜ q 500 500 m ˜ g n/s = 0 m 2 21. This is 200 /m h . ˜ q χ 100 200 m = 0 50 2 in which case we h 20 5 30. Right panel: the T 100 ) for the sum of the χ  1, Χ . ∼ 5.9 m = 40 GeV, and 50 = 1 χ m/T ˜ g m 119. Correspondingly, the . − /m ˜ g . Details of the evolutions of the = 40 GeV and ˜ q 20 = 0 m χ m /T 2 χ ≡ h m χ m 10 m − ˜ g 5 11 17 23 29 35 - m

- - - - - for the same representative values

10

10 10 10 10 10

 ˜ g ≡ s n , since then squark-Bino coannihilations should also ˜ g m /m = 7 TeV, ∆ ˜ q /m ˜ χ q – 13 – m m m ). The dashed line exhibits the naive thermal equilibrium are indistinguishable from the lines in the left panel. 500 1 5.9 = 7 TeV, ∆ 200 χ m T = 40 GeV. The left panel is for 100  χ Χ 30, as expected from a freeze-out calculation. In the right panel m m 50 − & ˜ g m for the representative case ≡ 20 m/T /T shows the effect of varying m χ 2 , the blue line shows the evolution of the gluino abundance, the red line that m 10 . Left panel: the evolution of the total supersymmetric particle abundance 1 compares a naive calculation using a single Boltzmann equation for the total relic , the relic dark matter density using the single Boltzmann equation is Ω 1 ˜ q 5 29 35 11 17 23 ------

m Figure

10 10 10 10 10 10

) for the gluino (blue lines), Bino (red lines), gluino-gluino bound-state (green lines) and sum

We do not show results for smaller values of  5 s n 5.4 7 TeV and ∆ find that the relic colddue dark to matter the density fact is that higher at than a previously: low squark Ω to gluino massbe ratio, taken there into is account. a cancellation among the corresponding naive thermal equilibriumand abundances. With the statedand choices the of full setblack of lines three in the Boltzmann right equations panel of yields figure Ω the Boltzmann distribution ifdeparture thermal for equilibrium wereof maintained, figure and weof see the a Bino clear black abundance, line the that of green theevolutions line of sum the that of gluon, the Bino of and gluino bound-state the and abundances. Bino gluino-gluino The densities. dashed bound lines The again states, inset show the shows and details the of the Figure abundance (left panel) with agluino, treatment Bino of the and three gluino-gluinofor coupled bound Boltzmann the equations state representative for abundances case the The (right dashed panel). line in These the results left are panel shows the total relic abundance as would be given by of the Bino and gluinoabundances densities are (black shown lines) in asabundances. functions the of inset. The dashed lines again exhibit the6.1 naive thermal equilibrium Bino LSP function of 10 using the single Boltzmann equationabundance, ( and the solidneutralino and line gluino shows densities, exhibiting theseparate the evolutions numerical familiar of freeze-out solution the when abundances of (solid( equation lines) using ( the full set of equations given by eqs. ( Figure 1 JHEP09(2015)127 , , ˜ 2 g < m 13 14 11 12 h - - - - 2 10 10 10 10 χ /m h ˜ q χ 500 500 Ω m = 120. In < 200 ˜ g 100 200 /m 50 = 150, we find ˜ q 1151 for our nominal . . The horizontal ˜ g m 20 ˜ g T 100 m as well as the very  1 (left), 120 (right). . /m Χ /m ˜ g ˜ q . ˜ q m ˜ g = 1 m 50 m /m ˜ g ˜ q . The effect of including , where /m χ ˜ q assuming the same gluino- 2 m /m ˜ m q 5 m ]) for a selection of values of for fixed model parameters, so 20 m 10 . In general, there is a shallow 2 49 ) plane where 0 × h 1 m χ 0 10 . ∆ at small extend to even larger values of ∆ , 2 χ 4 5 = 1 h 11 17 23 29 35 - 40, and 120 GeV (black, red, and blue

- - - - - 0. In both panels, the insets show details m ] the Sommerfeld enhancement causes a

χ ,

. 2 10

10 10 10 10 10

 h s n 28 χ , = 6 = 0 2 16 χ 13 14 12 h - - - – 14 – χ 10 10 10 m , but for the choices 1 − 500 channel leading to a smaller gluino annihilation cross 500 50 whose location depends on ∆ 100. In between there is a plateau with lower Ω ˜ g s bands in figure = 10 shown in figure ∼ m bands, calculated including the Sommerfeld factor, ap- 200 & 2 ˜ g ˜ g 2 ˜ g h 100 for fixed values of the model parameters, so long as the ≡ h χ 200 50 χ 2 /m /m /m and extend to larger values of = 200 we find Ω ˜ q m ˜ h q ˜ q ]. χ ˜ 20 g m m m T m 49 display bands in the ( 100  , as shown in the right panel of figure Χ /m ˜ g 4 ˜ q and the decoupling of the gluino coannihilations at high range for the Planck determination of the cold dark matter density m 50 m ˜ g /m σ 0014 [ ˜ q . , the relic neutralino density as a function of around at high 0 m /m 3 2 ˜ q 2 ± h h m 20 χ χ = 7 TeV, and ∆ below and above the current central value [ 1193 χ the relic density grows very sharply: for example, for . σ . As in the right panel of figure 10 ˜ g m = 0 , as calculated in various dynamical approximations. The red bands were calcu- 5 ˜ channel annihilations with the g = 780, and for /m 29 35 11 17 23 2 -

- - - - - ˜ q

The panels of figure In order to summarize the effects of both the cancellations in the annihilation cross h u 2 10

10 10 10 10 10

 /m h m s n ˜ q 1235 (3 χ Figure 2 . gluino-Bino conversion rates areCorrespondingly, large the enough orange Ω thatpear the at gluino larger coannihilation values isbound-state of effective. effects is ∆ to suppressthat further the the corresponding value of black Ω Ω 0 m lated dropping both thestate Sommerfeld formation. enhancement factor As and wassignificant the already suppression effect noted of of in Ω [ gluino bound- rapid rise in Ω as exemplified by theminimum case in Ω band indicates the 3- of Ω section at low we show in figure value of lines, respectively). We see clearly the rise in Ω of the evolutions ofof the gluino, BinoΩ and bound-state abundances.Bino At mass difference still of larger 40 GeV. values conversion These to large keep numbers pace reflect with the failure the of Hubble gluino-neutralino expansion for large and section and hence afor larger large relic values density. of Thethis case, results we also find a change, much even larger more value of significantly, Ω JHEP09(2015)127 -channel channels u ], and take u 50 = 7 TeV and that is typical χ ˜ g - and and t m t βm 500 -, s for range for the Planck ˜ g ]. σ 49 /m ˜ q is due to the decoupling of m ˜ g 0014 [ 1, where the . . 0 100 /m ˜ q ± = 1 m ], and they should be relevant also to the ˜ g 50 28 1193 . Ž /m g ˜ q = 0 m m as a function of  2 2 Ž q h h χ m – 15 – 10 2 to 6.4 TeV. In this case, the numerical effects . 6 is due to the cancellations between the ∼ 5 is for the case ˜ g 6 (coloured purple) the bands that would be found if the χ 4 4 /m m ˜ q 40, and 120 GeV (from bottom to top, black, red, and blue lines, m , = 0 χ ] in the context of a hidden-sector stau that excited bound states can at most 1 m 51 5 1 appearing in the Sommerfeld enhancement factor at a scale 1.6 in the thermally-averaged bound-state formation cross section, compared to

-channel contributions to the gluino annihilation cross section. Here we

0.5 0.1

− s

s ], and we expect a similar shift for our black bands. However, these two additional

0.05

∼ ˜ g α Χ W

h 28

10%, based on a comparison of the light green and dark green bands in the lower right m 2 ∼ ≡ m . The relic cold dark matter density Ω . We also show in figure in evaluating the bound-state effects. The full QCD potential and the thermal mass of the gluon χ s 3, which is comparable to the thermal velocities of the gluinos at the freeze-out temperature. Because α . m The upper left panel of figure We evaluate the 6 = 0 bands inward by panel of figure 2effects in could [ be compensatedthat by it contributions from was estimated excitedintroduce bound in [ a states that factor we of that do obtained not by include. considering only Wevalue. the note We ground therefore state, plot but purple the color bands charge with in an the uncertainty gluino of case a may factor change this 2 to allow for these uncertainties. of the momentum transferβ of the soft-gluon exchangesthe responsible for Sommerfeld the enhancement Sommerfeld issame effect a [ precursor towere the considered formation in computing of thecalculation bound Sommerfeld of states, enhancement the in for [ bound-state simplicity effects. we take We estimate the that these effects could result in a shift of our orange bound-state effects extends to of the Sommerfeld enhancement areboth similar effects to those are ofdark considerably gluino matter larger bound-state density formation, than represented and the by the current breadths observational of uncertainties the in bands. the The purple band, which bound-state formation rate were a factorhigher-order 2 QCD larger or than other our effects. calculations, as might arise from partially cancel the see that the black band calculated including both the Sommerfeld enhancement and gluino diagrams for gluinothe pair gluino annihilation, and and neutralinodetermination the densities. of rise the cold The at dark horizontal large matter band density indicates of the Ω 3- and Figure 3 the choics ∆ respectively). The rise at small JHEP09(2015)127 1 < . 10 000 10 000 = 1 ˜ g 1151 . /m ˜ q m 8000 8000 D D 10 120 = 6000 6000 = Ž g Ž g GeV GeV @ @ m m  Χ Χ  Ž q Ž q m m m m 4000 4000 2000 2000 0 0

50 50

200 150 100 200 150 100

Χ Χ

- - g g D @ D @ GeV GeV m m m m Ž Ž – 16 – 10 000 10 000 ) planes for a Bino LSP, exhibiting bands where 0 χ 8000 8000 m − ˜ g D D 50 m 1.1 = = 6000 6000 Ž ≡ g Ž g GeV GeV @ @ m m m  Χ Χ  Ž q Ž q ∆ m m , m m below and above the current central value), for different values of χ 4000 4000 σ m 1235 (3 . The ( 2000 2000 . 0 < 0 0

50 50

2

100 200 150 100 200 150

Χ Χ

- - g g D @ D

@ h GeV GeV m m m m Ž Ž χ with both the Sommerfeldand allowing enhancement for factor the andcalculations possibility gluino (purple that bands). bound-state the formation bound-state (black formation bands), rate is a factor 2 larger than our Figure 4 Ω (upper left), 10 (upperwithout right), the 50 Sommerfeld (lower enhancementthe left) factor Sommerfeld and and 120 gluino enhancement bound-state (lower factor right). formation but (red These without bands), results gluino with are bound-state calculated formation (orange bands), JHEP09(2015)127 , 2 h χ 10 000 10 000 ranges for 8000 8000 σ 2 to 7.5 TeV. . 7 D D ∼ 10 120 χ 8 (9) TeV. On the = 6000 6000 = Ž ) are quite different. g GeV GeV Ž m g @ @ ∼ 4 m = 50 (lower left panel m  Χ Χ  Ž χ q Ž q ˜ g m m m m m 4000 4000 /m 1235. ˜ q . 0 m < 2 h 2000 2000 χ ), the effect of bound-state formation Ω 4

. As in that figure, the calculations assume

<

0.10 0.08 0.06 0.04 0.02 0.00 0.14 0.12 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

Χ Χ 4 W W h h

2 2 1151 . – 17 – 10 000 10 000 band 0 σ = 120 (lower right panel of figure 3 TeV allowing for a factor 2 uncertainty in the bound- . 8000 8000 ˜ g 6 ∼ /m D D ˜ q 1 (upper left), 10 (upper right), 50 (lower left) and 120 (lower right). . 50 1.1 m = = 6000 6000 Ž g Ž = 1 g GeV GeV @ @ ˜ g m 1 TeV ( m  . Χ Χ  Ž q 6 Ž q /m ˜ q m m m = 10 (upper right panel of figure m ∼ m ˜ g 4000 4000 /m ˜ q displays these effects differently, exhibiting the positions of the endpoints m 5 ), where the black and purple bands also extend to ) of the gluino coannihilation strips as functions of the assumed value of Ω 2000 2000 χ . The locations of the endpoints of the gluino coannihilation strips for different values of 4 m 8 (9) TeV. These trends are also seen in the case

, using the same colour conventions as in figure =

Figure 0.08 0.06 0.04 0.02 0.00 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.14 0.12 0.10 ∼

2 Χ Χ W W h

h ˜ g

h

χ 2 2 χ m ( again from calculations with neither(red the lines), Sommerfeld with enhancement the nor Sommerfeld gluinowith enhancement bound but both states without the effects bound states included (orange (black lines), lines), and allowing for a factor 2 uncertainty in the The Sommerfeld effect is muchslightly larger larger than than the the bound-statethe widths effect cold of though dark the the matter coloured latter density.extending is bands Also, only still corresponding the to to allowed range thestate of 3- effects). the LSP mass is greatly reduced, from higher-order QCD, excited boundIn states, the etc., case extends tois larger somewhat smaller thanm the Sommerfeld effect, andof the figure black (purple) bandother hand, extends the to results for Ω the different values The horizontal green bands show the 3- includes an allowance of a factor 2 uncertainty in the bound-state effects, as might arise Figure 5 JHEP09(2015)127 χ is = 1, . ˜ g m χ = 10 band : the m /m ˜ g ˜ = 1 g ˜ − q σ ˜ g ˜ g m /m /m m ˜ q ˜ q /m = ˜ m q m 1 (upper left) m . m = 1 ˜ g (50) GeV and the freeze- 500 O is due to the cancellations /m 50% for ˜ q ˜ g would be relevant if the in conventional Big Bang ∼ m 2 2 /m 0042, and the brown and red h h . ˜ q χ χ 0 by m ± 2 100 h , as compared to the χ 2 h 1193 50 . χ Χ 15, respectively. The band and contours = 0 . m at the endpoint of the coannihilation strip  2 Ž q h χ χ m m 10 – 18 – . The drop at small 05 and 0 ˜ g . , the smallest value of 0042 (green band), as a function of . 0 = 50 (lower left) leads to a marginally smaller value 4 /m 5 ˜ q = 0 ˜ g ± 2 m h /m 5 TeV the bound-state binding energy is ˜ χ 15, respectively. q = 10 to 50. . the value of 1193 ∼ ˜ g m . = 120 (lower right) leads to significantly lower values of ˜ g for any fixed value of Ω 6 ˜ g m /m χ . The effect of including bound-state effects is to increase the 1 ˜ q = 0 2 m /m h at the endpoint of the gluino coannihilation strip when ∆ m 2 ˜ q 05 and 0 χ . 4000 2000 8000 6000

h χ

10 000

m χ -channel diagrams for gluino pair annihilation, and that at large

Χ m D @ GeV m 1235. We recall that lower values of Ω u = 0 . 0 2 h χ 20% for < - and t 2 ∼ compatible with the measured value of Ω h -, As was also seen in figure χ s range of the dark matter density: Ω χ = 0 and Ω Ω 7 m σ . The value of m < , and the choice χ Finally, we show in figure It is worth recalling that for 7 m 1151 . brown and red contours are for Ω out temperature in conventional Bigpaper Bang of cosmology standard is cosmologicaldark hundreds evolution matter of is scenarios GeV, rather where so different the the from freeze-out assumption that temperature made made may in in be this more in conventional the thermal GeV range. for any fixed valuerange of Ω of decreasing to when ∆ component, and parameter choices yielding highercosmology values (as of assumed Ω here)evolution. could be relevant in modelsleads to with smaller non-standard values cosmological of case (upper right). The choice of bound-state effects (purple lines).0 The horizontal greenLSP provides bands only again part show of the the 3- dark matter density, e.g., if there is also a significant axion Figure 6 0 in the Bino LSP case,between as the a function of due to the decouplingcurrent of 3- the gluinocontours and are neutralino for densities. Ω The green band corresponds to the JHEP09(2015)127 10 000 at the . The 2 3 h χ 3 TeV, in the . The reason ∼ 8000 100, reflecting χ χ m m & D ˜ g 10 , which has a shallow /m 7 TeV. = χ ˜ q 6000 Ž g shows how Ω GeV ∼ m displays the gluino-Wino m @ ], namely 7 m χ 7  Χ 27 ), Wino-Wino annihilations Ž q = 10, using the same colour m χ ˜ g m m m 4000 /m − for the case of a Wino LSP, with ˜ q ˜ g . We see that the black line crosses 6 m m 7 0042 and the brown and red contours 0042 for . . 0 0 , but for a Wino LSP. ± ± 5 2000 0042 for . and 1193 1193 0 . .

– 19 –

4

0.10 0.08 0.06 0.04 0.02 0.00 0.14 0.12 ± Χ W h 9 TeV when allowing for a factor 2 uncertainty in

The right panel of figure = 0

= 0 . As previously, the colours of the lines correspond 2 8 2 2 χ ∼ conversion. We conclude that, within the framework h h ]. We have included an estimate of the Sommerfeld enhancement m 1193 χ χ . χ 55 10 000 – is below the maximum value of − 3 TeV. = 0 52 ˜ g g . As in figure is the analogue of figure 2 ∼ < h 8 = 10 to 50, and falls sharply when /m χ χ 8000 ˜ q ˜ g m m D /m 10 ˜ q Figure 7 8 TeV (rising to = 6000 Ž g m GeV 7 TeV. Note that the curves appear to diverge at low m @ .  Χ Ž q χ ∼ m m m χ 4000 m 2000 The left panel of figure 0

The Sommerfeld enhancements in the calculations of the thermal relic abundance of a Wino- or Higgsino-

50

200 150 100

8

Χ

- g D @ GeV m m Ž like LSP were discussed infactor detail for in [ the Wino-Winolimit annihilations, where the and effect wefor of get the gluino a Higgsino-Higgsino coannihilation annihilations, similar isas since result absent. shown its as in effect We [ the do is literature. not much milder include in the the Sommerfeld relic enhancement abundance calculations, endpoints of the stripsto varies the with colours ofthe the horizontal strips green in band the where left Ω panel of figure the green band corresponding to Ω rate enhanced byextends a to factor 2).is that We even see inare that the strong in absence enough to of this suppressother coannihilation case the experiments (large relic the when density black below the coannihilation density strip indicated by Planck and We now consider the case ofcoannihilation a Wino strips LSP. The for left Ω panelcodings of figure as for thebound Bino states, case orange including (redstate the with effects, QCD black neither with Sommerfeld the both enhancement Sommerfeld effects but enhancement included, again and nor no purple gluino bound- with the bound-state formation maximum around the effect of astudied breakdown here, in ˜ the bound-state formation rate) in the Bino6.2 LSP case. Wino LSP exhibit the inverse ofneutralino the mass behaviour at of low the relic density seen previously in figure JHEP09(2015)127 1000 and 7 TeV β 500 and due ∼ , similar contours ˜ g χ 2 8 h m is due to the χ /m ˜ q χ 100 = 10 using the m m at the endpoints Χ ˜ g 50 2 m . In this case, when h displays the gluino-  300 which represents χ /m Ž χ q ˜ 9 q & m m m ˜ g 10 /m ˜ q 5 m , is similar to the Bino case. As to lower values of 0042 and ˜ g . . As in the case of the Wino, the shows how Ω ˜ 2 g 0 h 9 1 /m χ ˜ ± q /m ˜ q

8000 6000 4000 2000

m

100, with the drops in the Ω

10 000

Χ m D @ GeV m . 1193 . ˜ g is within the preferred range for 2 /m – 20 – h 1000 = 0 ˜ q = 10. Our results are very weakly dependent on χ = 0. The Higgsino couplings depend on tan 2 300 representing the decoupling limit. In this case, m 500 β h m χ . & 6 TeV. As seen in the right panel of figure 100. The percentage increase in the allowed range of ˜ g . The black line crosses the horizontal green band where ∼ . /m 9 ˜ χ g ˜ q 100 3 TeV. m m conversion at high /m Χ ∼ ˜ 50 q χ χ m , but for a Wino LSP (left panel) and a Higgsino LSP (right panel). m  Ž q m − 6 15. We see that Ω . m again being due to cross section cancellations at low g . χ 0042 for conversion. The curve hits a plateau for . 10 0 m χ for the Bino case, the fall in the Ω ± 5 , with the colours of the lines corresponding again to the colours of the − 6 χ g are found for a range 5 05 and 0 . m . As in figure χ 1193 . m 2 TeV, Higgsino-Hiiggsino annihilations are sufficient to reduce the relic density = 0 . 1 2 1 6 TeV at the endpoint where ∆ = 0 h

2000 6000 4000 8000 χ

∼

2 due to bound-state effects, as a function of <

∼

10 000

Χ h D @ Figure 8 GeV m χ χ χ χ to the breakdown ofcurves ˜ drop to a plateau for Ω values of to lower values of in our calculations wethis have choice. taken Once tan againm we see a divergence ofbelow the the contours Planck at density. low varies The with right panelstrips of in figure the left panel of figure same colour codings asenhancement nor for gluino bound the states, orange Binobut including and again the QCD Wino no Sommerfeld enhancement bound-state casesbound-state formation (red effects, rate with black enhanced with by neitherm a both the factor 2). effects Sommerfeld In included, this and case purple the black with strip the extends to the decoupling limit at 6.3 Higgsino LSP We now consider the caseHiggsino of a coannihilation Higgsino strips LSP. The for left panel Ω of figure to Ω over a broad rangem 5 shown in figure breakdown of ˜ JHEP09(2015)127 that 10 000 χ m 8000 8 TeV is weakly 8 TeV, increasing ∼ D ∼ 10 6000 = Ž g GeV @ m  Χ Ž q ], namely that coannihilations m m 4000 56 allowed in the Wino and Higgsino cases, 2000 50, but is substantially reduced for either , but for a Higgsino LSP. χ 5 1 TeV if the LSP is a Wino, and by a further ∼ < m ˜ g ∼

and – 21 –

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 /m the relic abundance by coupling ‘parasitic’ degrees

Χ 4 W ˜ h q 2 m ∼ < 2 TeV. The percentage increase in the allowed range of . 1 . In all cases, the percentage increase in the allowed range 10 000 increase ˜ g ∼ /m is . As in figure ˜ q 8000 χ m m D 10 6000 = Ž g Figure 9 GeV @ m  Χ Ž q m m 4000 2000 due to bound-state effects may be as large as 50%. 9 TeV if we allow for a factor 2 increase in the bound-state formation rate above our χ We are loath to claim that our upper limit on the LSP is absolute, but we do note that In the Bino LSP case, we find that at the endpoint the LSP mass The decreases in the maximum values of 0 due to bound-state effects is again similar to the Bino case.

50

100 200 150

∼

Χ m

1 TeV if it is a neutral Higgsino. The upper limit on the LSP mass of - g D @ χ GeV m m Ž sensitive to the squark masssmaller or for larger 10 values of of it is substantially higher than what is possible along the stop coannihilation strip, reflecting As we show,difference these along can the coannihilation affect strip, significantly and also the the preferred positionto range of the of endpoint. thecalculations. gluino-LSP These values mass are decreased∼ by We have studied inwith this the paper gluino, MSSM exploring scenarios thecoannihilation characteristics in strip and which in locating the the the cases LSP endpointsHiggsino. where of is Important the the ingredients (almost) LSP gluino-LSP in degenerate is ourannihilation the analysis rates, are Bino, the gluino-gluino the Sommerfeld neutral bound-state enhancement Wino formation of or gluino and a gluino-neutralino neutral conversion. abundance. This effect is compensatedwe by find the in decreases the in the Wino maximum and values Higgsino of cases. 7 Summary compared to the Bino case,may, under are some due circumstances, to theof effect noted freedom. in In [ thefreedom, Bino linked (Wino) by (Higgsino) coannihilation case, to there the are gluinos, 2 that (6) contribute (8) incrementally electroweak to degrees the of relic the asymptotic value of m JHEP09(2015)127 , s ) = (A.1) s ( 12 σ 2 ) by s |T | ( ) w 4 1, where many more in the centre-of-mass p ]. − → 61 CM 3 θ ˜ g p ]. − 4 in [ (summed over final spins and /m 57 ˜ 2 q , 2 p W/ m 12 3 + 4 as is necessary for solving + |T | 1 → is due to the breakdown of gluino- p ( ˜ g 4 δ . 4 /m ) 2 ˜ q is due to cancellations in the annihilation π m ˜ g (2 |T | ) is given in terms of 4 s /m ( CM E ˜ q – 22 – 4 θ 2 12 p 3 ], which is written as and the scattering angle m 3 σ ) d 60 s cos π – in an efficient manner, extending the approach used d (2 58 5 , expressed as functions of the Mandelstam variables 1 3 +1 for the process 1 + 2 B − E 3 Z 2 ) in [ i p s 3 ) ( 3 ) 2 rel s d / w π ( v 1 3 in terms of s (2 12 p ]. Next we define 2 σ h π 57 Z 1 |T | 1 4 32 = ≡ ) s 9 ( ). w s ( More details of our approach can be found in [ 1 ) is also the same as s p 1 ( . 2 / w 1 /s We start with the squared transition matrix elements Nevertheless, our analysis does show that a large mass reach to at least 8 TeV will be ) . We then express Our SSARD 9 s u ( , The total annihilationw cross section frame, as described in [ the Boltzmann equations in section in colors, averaged over initialwhich spins are and given colors) here for int appendix the coannihilation processes of interest, A The procedure for obtainingIn this a appendix, thermally we first averaged recallweighted cross our cross-sections procedure section for computing the thermal-averaged velocity- by the London Centre forResearch Terauniverse Council Studies via (LCTS), the Advanced using Investigator fundingthe Grant 267352 from research and the grant from European the ST/J002798/1. U.K.Research The STFC Council via work Advanced Investigator of Grant 267352. F.L.in was The part work also by of supported K.A.O. DOE was by grant supported the DE-SC0011842 European at the University of Minnesota. Acknowledgments The authors would like toHua-Sheng thank Shao Brian for Batell, Mathias very Garny, helpful Matthew discussions. McCullough and The work of J.E. was supported in part coannihilation processes would comesupersymmetric into models. play, as might also be the casenecessary in to explore non-minimal conclusively theMSSM possibility and of a supersymmetric conventional dark cosmological matter framework. within the charge. However, these annihilationnotably the rates squarks also in depend theupper gluino on limit NLSP the on case masses studied thematrix of here. LSP elements, other As mass whilst have sparticles, for shown, the small theLSP decrease decrease conversion. at in However, large we have not studied the limit the larger annihilation rates that are possible for the gluino because of its larger colour JHEP09(2015)127 , -, B s d and A (B.1) ). (A.2) ), and ¯ B u ¯ q A A.1 → q 5.13 ], using the − j → in powers of ] for the case . 57 gχ ) 0 i 2 u 62 -channel gluino- , gχ u |T | ×T , ˜ 58 t B T ¯ q at two different values A + q - and 2 u t → |T | ×T , ˜ g s
g T 2 , while the zeroth- and second- , ˜ − + CM x t gg θ O is used to average over initial colors. ×T → s + ˜ g ini T ) of the thermal-averaged cross-sections 1 g c − + x 2.3 u , so we do not discuss it in detail here. One 2 12 3 for three generations, the neutralino index b , waves, respectively. We can therefore evaluate ×T m 2 u for each final state. When the conversion rates p , + – 23 – T 6= ) numerically to obtain the relic density. 2. B , 12 1 ) simply by evaluating + a = 1 s and t m ( ) by following the prescription given in [ 5.10 = 1 s 2 = w ×T j i t A, B T by performing the sum over initial states as in ( rel < m v + 1 eff 12 in the expansion ( s b σ m b h ×T for each of these processes, separating the contributions from -channel gluon-exchange diagram, and s s and ] for the case 2 T and ( eff a 61 |T | 2 is needed when performing the momentum integration in ( ini , a c / 59 (assuming = takes the form 2 2 /T , where the indices 1 -wave contributions to |T | There is an B p ¯ |T | m d . 2 (co)annihilation processes relevant to the gluino-neutralino (and, in the Wino A . The odd powers vanish upon integration over , and in [ 4 and the chargino index . ≡ 1 2 -channel diagrams. In the following expressions, final spins and colors are summed u CM u → x gg m θ - and /m , ..., → processes Here we list the The proper procedure for thermal averaging has been discussed in [ The usual partial-wave expansion can be obtained by expanding ) s = → s + j ( 1 = 1 ˜ g 1 - and ˜ ˜ which we do not list separately. g exchange diagrams. We note that,an because extra there factor are of two identical 1 gluons in the final state, We note that there is also the charge-conjugated process for the chargino, ˜ i t over, and initial spins areTherefore, averaged over. A factor The 2 and Higgsino cases, gluino-charginos)gχ system are ˜ we then integrate the rate equation ( B List of the transition amplitudes for the gluino (co)annihilation where transition amplitudes listed in appendix are large compared to theeffective Hubble coefficients rate, these amplitudes can be used to compute the total m finds the coefficients for the processes of interest: p order terms correspond to thethe usual of cos JHEP09(2015)127 , . ˜ t u t 2 θ ˜ t ↔ m ×T (B.2) t t , = T ,  -channel 6 2 ˜ u U 6 up-type  ) 3 u ) m × u − = 0 for them). t − - and are also defined and  t t u 3 6 1 + ( ) ˜ t − u ˜ 4 g 1 ×T m ˜ s D ) + ( )] m + t T u t m = , − 5 = + 8 + ˜ U u t          t ˜ 2 g ( + 2( 1 2 1 2 2 1 2 m m ˜ ˜ t t ˜ ˜
c c ˜ ˜ s u u ) , ×T 2 2 + 4( s u u ˜ c +          T s 3 [ + can be different even if we restrict m -channel gluon-exchange diagram,          s ˜ 6 g t = + 2 s , respectively, by exchanging ˜ t B t = ˜ t 2 θ + s m 4( , t θ 4
− ˜ o U ×T u sin s ×T t + 2 and cos
 s m T
  + − T 2 , A 2 t tu ˜ t ) 1 ˜ t u ˜ , because the quark CKM matrix is involved c θ θ ) + 2 u − − and B u + m ¯ also has an ˜ ˜ 2 g 4 g d + t 2 These three processes all have . The mass eigenvalues t – 24 – A t = m m b ˜ B , + 3 u ¯ θ 4 q ×T 6( 3 . t  s ˜ A ( U 2  B ) q T → 2 − ) ¯ + 2 u d u m s u )  j + A , → u − 0 0 0 10 0 0 0 0 sin 0 1 0 0 00 0 1 0 00 0 0 0 00 cos 1 0 0 0 0 0 0 − ˜ 2 g su 2 u − ˜ ˜ 2 g u ˜ g t gχ t 2 s m (          g ( + 2 m → m s α + 2 n , is introduced for the down-type squarks, by changing  t 2 = 2 − 2 ( + j ˜ = π D s s     2 2 s t t t  2          2 s Z 2 s ˜ gχ , which relates the interaction eigenstates and mass eigenstates α ˜  U 2 α ˜ L R − − − L U R L R , tu − 2 s 2 ˜ t ˜ ˜ t c ˜ 2304 ˜ π c ˜ u u m are related to ˜ ˜ ˜ 2 g 2 g 2 g B 2 Z π α − ¯ , q s 2 u m m m          1  ˜ 8 g A 2 π ˜ u    ˜ q 4 g s 576 2304 m , 2 2 2 ×T m m 4 s s s . In the case of ˜ u → 1 u T 1152 64 = − − + − − 0 i 1 ×T ˜ = = = = = U t ˜ gχ and T t t s u , m ini u is the stop left-right mixing angle. The mass eigenvalues are correspondingly B c ¯ q ×T ×T ×T ˜ t ×T t s and s t A θ ×T T q T T T s u T We find → ×T ˜ u g ˜ where defined as A similar mixing matrix, to the sbottom left-rightsimilarly. mixing angle, the up-type and down-typesquark squarks. mixing Therefore, matrix, itof is the convenient up-type to squarks write as a follows: 6 whereas the other twoApart processes from do the not (hence couplings,T these three processesin the have chargino-quark-squark the vertex, the same indices to structures the as case for of no generation mixing with only left-right mixing in the third generation for and in the corresponding expressions. g squark-exchange diagrams, and ˜ JHEP09(2015)127 , i ) , is ) u  u io 2 u,c,t ) u ↔ B, p m ( ×T t
io + u u R ∗ ) = B, p 2 ) T , g ( t 4 3 , u R  2 , + g m B, p i 1 A, q ( 1 4 3
∗ ( U u R t u L m  m g m f p ) )
− . t 2 3 + 2 +3) ) + 2 − 4 1 m A . ˜ A, p B, p , f ( 2 4 ( ( U, D u m + Z t u R L m
 B, p  f g
 2 2 = ( ∗ ↔ ∗
) + u L ) m ˜ t f t + f Ap g ) + 2 2 2 ∗ Z u
4 − )  m B, q u A, q 2 2 3
( + m −

( in the indices, and the ( ∗ 1 1 t R t t − ∗ m R ) ˜ A, q f g Ap f ( 2 4 m m  ) R ) − + Z t R take the following forms:

m ˜ s 2 1 f f Ap B, q ( ) u ( ) + ) + , the index + 2 3 Z + B, p m A, p t L and in both the indices and the Mandelstam ( ˜ (  2 2 1 D g m

t t ×T R 4 ) R p u A, p L p t by exchanging B, p g m B, p f ( ˜ ( U, ( T m  t t L + 2 u +3) h h u L ↔ L q +
+ +3) , f 2 t B, p ∗ ∗ g t = g
A ˜ t ∗ A f ( ∗ ) ) ˜ ( f 2 ×T ( ) u

˜ ) m ) 3 f t R and s Z ) i u Z – 25 – g − h tu T 2 )

− − m u
and B, q B, q 1 s t − t  B, q A, q + ( ( m − u 2 ( ( t  t L t m R ∗ ×T + − 2 4 t t L B, p 2 L 1 ) g 2 g − m   t u ( ↔ g − f 2 3 s 2  2 m p t ) ) T u R 1 2 4 2 3 u 2 2 + m m g  ↔ m , + t B, q 2 q m 4 3 m m
n n 2 t 2 3 1 ( − ) ) A, p B, p u t R m . The index applies to all the 2 1 2 3 ( ( u, m + + m m − m ×T g t A t L L 2 t 2 3 − 2 2 ) u is related to m m f u + f g − T 2 s A, p in the indices and the Mandelstam variables. Again,  t ↔ A, p  m 4 1 m ( u 2 1 ( t m α t  2 1

u u u ×T R L − + 3 m t t    tt B, p m 2 s t 4 f f  = ( m c 2 2 2 1 ×T 2   2 s T ∗ α ∗ p + ( 2 ˜ t  s L p m − − 3 ) ) 2 f α 2 t m m h g π T h ) + and 2 3 2 4 2 s ) AB 3 ∗ m t ∗ m tu we find m ) t ) α δ + ) 2 c 4 m m ) in 2 A, q A, q 2 3 − 2 1 R B ( − ( π R m − AB π ¯ + + t q t t L t L and m 2 2 m δ by exchanging B, q , and B, q s  2 1 2 2 A f f 1 ↔ s  2 ( ( ( ˜ g t 3 q =1 m ↔ s  6 192 1 2 3 π t t L R m m m 2 L X m g m g =1 p,q  2 d,s,b ×T 6 L ( m m → t 6 =1 X = + + + + + 384 T m p × + p,q + ( × − X + − ˜ g 1 g = = = = = m 3 t t , s u 2 The couplings and masses involved in the above expressions are listed below. For all three processes, For ˜ , 1 ×T ×T ×T ×T D t s s t T T T where the ( applies to both the related to variables. T where m JHEP09(2015)127 ,  A 3 f T i2 N ,  + ∗
) , A 3 i2 f ∗ T N  ( − B ˜ , f Bp 3 f A , f Z T w ˜ ∗ f Ap  Q +  m Z t R p w 2
u L ) c c , B w √ B m , 3 f A +3) f θ ∗ 2 f p T , B ˜ f  m ( √ m 2 ˜ − Z +3) f 2 ˜ Ap f Bp tan ( ig  A ˜ B Z f ig ( Z f i1 . − , −  Z − , Q , − N p ∗ ˜ ,  f A p ) ˜ g f  3 U, D m ˜ p w f Ap 256 m m +3) θ = ) = Z ) = ) = ) = A ˜ f +3) ( 2 = – 26 – f g B = , = = Z ˜ f ( 2 tan ( ) A, p B, p A, p B, p , p , Z 3 ( ( ( ( ∗ √ w 32 2 4 u ∗ i u u  s u u L R L R θ ) uu 1 64  ) c − m m m f f g g 8 i1 p − w πα N θ p ======( tan ( +3)  ) = ) = ) = ) = 1 3 p ∗ tt t B tu ˜ ∗ ini +3) = f ( c w ) c , m m c , A tan ( ˜ m  f Z ( i1 ∗ m p A, p A, p w B, p B, p ˜ B  f ( ( Z ˜ 2 ( ( tu N f  f Bp t c m t ( L t , and the index t L p L R u R t R m √ Z Q 2 c f c g f g ˜ − A , D  , A B i1 f √ +3) ∗ f f 2 p , = Q N ˜ B ˜ g U,  f ( = 2 2 m m : 2 , 2 2 ˜ g g +3) : Z f ˜ , Ap B 0 i f p = Bp 2 2 √ ¯ q A  ˜ ˜ B ig ig , u g χ Z i f ( ˜ Z ¯ q f uu A √ √ 1 8 m Z − − c m i −  − − i m q A q = = = = = → ) = ) = ) = ) = ) = ) = ) = ) = → 1 2 p 0 i tt t ini c ˜ g m m c m gχ g A, p A, p A, p A, p B, p B, p B, p B, p ( ( ( ( ( ( ( ( t u t u L u t L L L R R u t R R For ˜ For ˜ f f g g f f g g where the index JHEP09(2015)127 , and ˜ U = ˜ f is the Standard 2 g , , ∗ ∗ ) ) i3 i4 ], and N N 63 )( )( β β . ! 2 0 v = sec( = csc(

, and ∗ ∗ ˜ ) ) i ≡ D 2 u R u R c c = H h ˜ f , = ( = ( – 27 – ! , , 1 ∗ ∗ 0 v , 1 3 ,

1 2 , s 2 3 t R t R −

1 2 − c c 8 πα = = i ≡ = = , . , , , . , , 1 = = B A B A = A B A B H B B B B D D D D U U U U h f f 3 3 f f u L u L T m m m m m m m Q c m Q c T tu c ======− , 4 3 3 4 t L t L A A A A A A : B B p 3 3 f f f f f f , f f c c B , m m m m = , 4 neutralino mixing matrix as defined in [ , , T T , ¯ Q Q d m m m m ˜ +3) U B p , Ap A p j + × A ˜ ˜ A ˜ , ˜ χ U D U D g Z U ( u uu coupling constant. For up-type quark final states, the index 1 8 m m m m Z − c m m , and the vacuum expectation values of the two Higgs doublets are defined as, → L 1 ======+ j ) = ) = /v 3 4 1 2 p p is the 4 tt t 2 u ini c gχ v m m m m c m m N A, p A, p ≡ ( ( t t For ˜ L R β f f For down-type quark final states, the index tan Model SU(2) where JHEP09(2015)127 B ¯ q , A that q (B.3) - and ˜ t g   ∗ → -, /m s  ˜ g ˜ q g ˜ U Cp m Z  (2014) 055 ∗ ) 2 chargino mixing 06 j1 , × V )) for the ˜ . On the other hand, ( 2   ˜ − g ˜ A.2 CB D -channel gluon-exchange Cp m JHEP s , Z TeV proton-proton collision K , 2 2 j1 2 are the 2  ig U  ˜ = 8 2 q ˜ 2 q TeV V in eq. ( s − AC m m a √ ∗ K ]. = 8 − + 2  and , only the ˜ s 2 g , p ˜ 2 g ˜ g ig √ , m U m m   +3) −   ∗ SPIRE C ! ˜ 2 s p U ˜ 2 g (  IN  ˜ Z D ˜ Cp m q πα ][ ˜ +3) U  Cp 8 Z 9 m C ˜ ∗ ∗ Z D ( ) )  = Z j2 – 28 – j2 C B w V V D D ( ( m 2 C m m w A w w U U j2 j2 √ m m m U m U 2 m 2 ) limit value 2 ) √ β √ β B CB CB ), which permits any use, distribution and reproduction in √ ¯ q ]. A arXiv:1405.7875 K K q ) ) [ , sec( csc( β β → ∗ ˜ g Search for squarks and gluinos with the ATLAS detector in final ˜ g  AC AC ]. SPIRE p a csc( sec( K IN K Search for new physics in the multijet and missing transverse 63 , 2 2 2 2 +3) ][ ∗ CC-BY 4.0 ig ig ig ig B ˜  D ( (2014) 176       Z ˜ This article is distributed under the terms of the Creative Commons D Bp  09 =1 =1 =1 =1 3 3 3 3 Z , the above expression approaches zero. This cancellation of the X X X X ˜ g C C C  − C matrix is the quark CKM matrix, and collaboration, m K collaboration, ) = ) = ) = ) = ) = ) = is the common squark mass. When JHEP → , ˜ q ˜ q m A, p A, p B, p B, p B, p B, p m ( ( ( ( ( ( arXiv:1402.4770 CMS momentum final state in proton-proton collisions at data [ ATLAS states with jets and missing transverse momentum using u u Finally, we give the s-wave result (i.e., the coefficient u t L L L R u t R R f g g f g g [2] [1] -channel contributions results in the feature of the plots at small values of where the any medium, provided the original author(s) and source areReferences credited. when u are commented upon in the main bodyOpen of Access. the text. Attribution License ( where diagram contributes, and the above expression is proportional to channel, in the limit ofleft-right a mixing common in squark mass the andthe squark massless text). mixing quarks, In matrices with this no (thethe limit, generation case same, the or considered and contributions in the from the result each of main of putting the body all six of the quark six flavor final quark states flavors together are is matrices as defined in [ JHEP09(2015)127 ] B Phys. (2013) ]. , (1983) ]. Phys. Phys. Rev. , Eur. Phys. , Phys. Rev. 50 , D 87 Phys. Lett. , SPIRE ]. , SPIRE IN IN hep-ph/0301106 ][ ]. [ ][ Supersymmetric SPIRE ]. IN ]. SPIRE ][ Phys. Rev. ]. IN , ]. Neutralino-stop SPIRE ][ Phys. Rev. Lett. (2003) 001 IN SPIRE , SPIRE (2007) 115005 ][ IN 04 SPIRE IN Compressed supersymmetry at ][ IN ][ arXiv:1409.2898 [ hep-ph/0701229 [ [ D 75 ]. ]. 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