JHEP03(2017)005 , )M ˜ Ga 6 , with in the 10 → a 2 − / a 1 Springer 3 Gev

Home , Warm dark matter

JHEP03(2017)005 , )m ˜ Ga 6 , with in the 10 → a 2 − / a 1 Springer 3 GeV. − m 16 , and a high March 1, 2017 a : February 2, 2017 December 3, 2016 ) and gravitinos and can be any- : : a 2 / 3 m , solving the gravitino 2 Published / a,b 10.1007/JHEP03(2017)005 3 Accepted Received and approaches 10 m doi: R T PQ . The free-streaming scale for V ˜ aa → anywhere between the keV and TeV ˜ , which is in the range of (10 Published for SISSA by 2 G / [email protected] 3 , depends on m GeV h/Z/γ PQ and Lawrence J. Hall V 16 + ˜ GeV, for any value of LSP the warm component arises from ˜ G c,d ˜ 10 16 G . A large overproduction of axinos (˜ and with mass R T ˜ G h/Z ) GeV. Detailed predictions are made for the lifetime of the + [email protected] a 16 , 10 , and from freeze-in at the TeV scale, is diluted by the late decay − R . T 10 3 Francesco D’Eramo can be as high as 10 1611.05028 R a,b T The Authors. c Supersymmetry Phenomenology

A new scheme for lightest supersymmetric particle (LSP) dark matter is in- , LSP the warm component arises from [email protected] a LSP). ˜ G ) from scattering at Department of Physics, University1156 of High California Street, Santa Santa Cruz, Cruz,Santa CA Cruz 95064, Institute U.S.A. for1156 Particle High Physics, Street, Santa Cruz, CA 95064,E-mail: U.S.A. Berkeley Center for Theoretical Physics,366 Department LeConte of Hall Physics, MC University 7300,Theoretical Physics of Berkeley, Group, California, CA Lawrence 94720, Berkeley1 U.S.A. National Cyclotron Laboratory, Rd., Berkeley, CA 94720, U.S.A. ˜ G b c d a Open Access Article funded by SCOAP contribution to dark matter is sub-dominant, except when Keywords: ArXiv ePrint: case of problem. The PQ symmetrywhere breaking in scale the range (10 neutralino LOSP decaying to ˜ over much of parameter space. For an axion misalignment angle of order unity, the axion of a saxion condensate that arisesmass from of inflation. order The two the lightestscales, TeV superpartners depending scale, are on ˜ and the mediation scaleboth of warm supersymmetry breaking. and Dark coldwhile matter components: contains for for ˜ the warm component is predicted to be of order 1 Mpc (and independent of Abstract: troduced and studied in theoriesreheat temperature of after TeV supersymmetry inflation, with( a QCD axion, Raymond T. Co, Gravitino or axino darktemperature matter with as reheat high as JHEP03(2017)005 6 9 16 GeV GeV 10 10 10 20 10 & 25 . R 36 R 35 37 T T 17 – 1 – 16 31 12 12 19 32 15 33 13 20 29 24 26 20 35 10 6 32 33 theory theory theory theory 0 + 0 + 33 2 B.1 Saxion decays B.2 Neutralino andB.3 chargino decays to Gravitino axinos decays B.4 Axino and neutralino decays to gravitinos A.1 Color anomaly A.2 Supersymmetric interactions A.3 SUSY breaking interactions 5.1 The DFSZ 5.2 The DFSZ 6.1 The DFSZ 6.2 The DFSZ 4.1 Warm dark4.2 matter from NLSP Displaced decays signals4.3 at colliders Axion dark radiation 3.2 UV production3.3 of axinos UV production3.4 of gravitinos Freeze-in production of gravitinos 2.1 High reheat2.2 temperature after inflation: Low reheat2.3 temperature after inflation: Field equations 3.1 Freeze-in production of axinos C Free streaming of warm DM component B Decay widths A Axion supermultiplet interactions 6 Results for low scale or “gauge” mediation 5 Results for high scale or “gravity” mediation 4 Axino and gravitino as the lightest superpartners 3 Axino and gravitino production Contents 1 Introduction 2 Saxion cosmology JHEP03(2017)005 ] 2 ) of , ]. 1 (1.2) (1.1) ], but 15 at the A.5 – 16 R ], where 13 T 6 , 5 ] that relaxes the 4 , 3 from gluino scattering [ R a T ], either through decays or 8 , ..., 7 ) colored fermion carrying PQ + . The superpotential cubic cou- ), instead of the axion decay constant weak scale supersymmetry and d PQ A term as well as a coupling of the V H F A.2 2 u µ θ both + AH . This axino production by the freeze-in ˜ a GeV a motivated possibility. ˜ a θ 2 PQ µ 12 V → √ µ χ q + defined in eq. ( – 2 – + ia 2 PQ d V + √ H ], namely most of the axinos are produced at temper- u s of order 10 ], analogous to UV production of gravitinos [ 10 , both play central roles in cosmology. term of the minimal supersymmetric standard model , = µH a 9 PQ 12 , µ A V = 11 , PQ breaking induces the DFSZ operator generated when the top quark is integrated out [ PQ W V leading to a light axion degree of freedom to zero. In this case dark matter could be axions produced by the ˜ G 1 , must be promoted to a superfield ¯ θ , a aG PQ , and the axino, ˜ V s there is at least one heavy (with mass of order , an IR contribution to axino production via scattering also arises from the . the heaviest colored fermion carrying PQ charge is the top quark, so the only 0 : : a model dependent parameter defined in appendix A 0 + µ source for axino production is the IR dominated freeze-in; charge, and thus we also haveoff UV quarks dominated production and at gluons. q In order to make this distinction sharper, we define two different types of theories: In this work, for reasons discussed below, we focus on DFSZ theories [ However, the cosmology of these two leading candidates for dark matter, LSPs and -violating phase In this work we use the PQ breaking scale 1 . These two quantities are connected by a color anomaly coefficient, as shown explicitly in eq. ( a In DFSZ supersymmetrized f appendix DFSZ atures around the TeV scale.a Depending large on abundance the of fermion axinos contentend of can of the also inflationary PQ be breaking reheating produced sector, [ in the UV, at the temperature DFSZ with pling is responsible for axino productioninverse in decays the of early charginos universe [ and(FI) neutralinos, mechanism ˜ is IR dominated [ the PQ symmetry(MSSM). forbids At the the scale axion supermultiplet with the MSSM Higgs superfields axions. The axion, and the saxion, broken at scale CP misalignment mechanism, with axions, is changed enormously in theories that have Perturbative theories with supersymmetrythe broken at hierarchy the problem, TeV even scalemasses if in are the they well-motivated region by do discovered at(LSP), not the LHC. completely cosmologically Dark solve matter produced could it,strong by be and CP the the problem lightest lead superpartner is freeze-out to elegantly solved mechanism. Higgs by introducing boson a On Peccei-Quinn the (PQ) symmetry other [ hand, the 1 Introduction JHEP03(2017)005 16 ∗ (left 0 10 M = 1 GeV = 10 GeV = 100 GeV = 100 MeV = 1 TeV, 3/2 3/2 m 3/2 µ 3/2 m , followed by m m is very large ˜ a 15 10 2 = → ) plane for four / PQ . The thick (thin) 2 R χ V R + M T ,T (GeV) = both for DFSZ 14 PQ DFSZ . For UV production in PQ 1 V 1 10 V µ M . ˜ a DM m GeV. Thus LSP dark matter 13 8 10 . 10 2 / overproduction 3 . m R 12 T 9 8 7 6 5 4 10

10 10 10 10 10 10

R ) ( GeV T (right panel), with + – 3 – or of order the cutoff of the field theory, 16 10 is very large so that axino freeze-in is significantly PQ = 1 GeV ]. This is illustrated in figure = 10 GeV V = 100 GeV = 100 MeV 3/2 17 3/2 PQ [ m 3/2 3/2 m V m m R 15 T 10 (left panel) DFSZ is either 0 0 I s (GeV) 14 DFSZ PQ 10 V (right panel) for gravitino LSP and other superpartners at the TeV = 6. Vertical lines correspond to axino freeze-in via decays ˜ + = 2. The axino mass is in the range µ DW DM 13 q N 10 overproduction . Contours yielding the observed dark matter abundance for a gravitino LSP without a is very low. However, if , whereas horizontal lines correspond to UV gravitino production at we fix R = 2 and + 12 ˜ T G a Saxion cosmology can greatly change this conclusion. Axion theories typically have In the absence of a saxion condensate, UV production of both axinos and gravitinos 4 6 5 8 7 9 10 β

10 10 10 10 10 10

R ) ( GeV T → ˜ a domain wall problem,inflation, which we and assume not is restoringsaxion solved it vev by breaking afterwards. is the zero.saxion, PQ We displacing the define symmetry During vev before the away inflation fromterm, saxion today’s supersymmetry the value. field breaking vacuum Depending so value on yields that the sign a today of potential the the quadratic for the is typically overproduced inand the absence of asuppressed, saxion the condensate, universe unless ismisalignment typically angle overclosed is by selected axions, by unless an a anthropic low requirement. value of the axion puts a very powerful boundpanel) on and DFSZ scale. Contours that yieldvalues the of DM the relic density gravitinotemperature are masses. shown after in Even inflation the for is ( a strongly gravitino bounded, with a weak scale mass, the reheat it is suppressed compared togravitino the mass, one the from decay decays ofbut and neutralinos we we can neglect find also it. this populate contribution gravitinos In sub-dominant by theories to the with FI the a mechanism ones low mentioned above. saxion condensate for DFSZ tan DFSZ a portions of the lines refer to a freeze-in contribution smaller (larger) than 50%. Figure 1 JHEP03(2017)005 ], R ∼ T 25 PQ V GeV [ 15 has a serious . 10 PQ PQ bosons, giving a ∼ V V the scale of grand Z , and of axinos pro- PQ or GeV–MeV for R V . T ∼ GeV to ˜ W ]. The freeze-out mecha- χ ˜ χ 12 Rs 24 , giving a saxion condensate and lower , T ˜ 10 G, ]. Hence we focus on DFSZ I ˜ s a R 23 ∼ 22 T ˜ a, ˜ a → s term is not forbidden by PQ symmetry GeV, when the saxion condensate never µ ; for example, 13 Rs , this saxion condensate comes to dominates low enough to suppress UV production, there 10 ]. In this paper we consider LSP possibility by T ∗ R – 4 – 3 26 . M T ∼ PQ I V s ∼ I and allowing much higher s 1 GeV or 13 10 as the reheat temperatures after inflation and saxion decays, respectively. GeV, the entropy is released after the axion condensate starts to ] axion theories the MSSM & Rs 13 T = 0 requires a special symmetry structure that we do not consider in this paper. I 10 ]. s 21 I , s needed for axion dark matter is raised from and 26 If > a 20 f R 2 T ]. GeV. However, as we have already mentioned, this case of low PQ ) GeV, respectively. The decay of the saxion condensate has crucial implications V 16 12 19 , 10 10 the constraints of figure If kinematically allowed, thepartners, saxion and condensate therefore leads LSPs, to via a such late decays production of super- value of suggesting the possibility thattogether PQ [ and grand unified gaugeThe symmetries large are abundances broken ofduced gravitinos by and freeze-in axinos at the produced TeV at scale, are diluted by saxion decays, greatly weakening nism can only give thethat observed dark are matter then abundance diluted. ifTeV scale. it first This overproduces could LSPs happen byFor raising superpartners welloscillate, above diluting the misalignment axions. For a misalignment angle of order unity, the The LSP abundance from freeze-out is greatly diluted [ 18 − [ ∼ In our scheme the dominant saxion decay is to pairs of Higgs, In KSVZ [ The possibility We define • • • • 13 2 3 PQ PQ production mechanisms for dark matter?unification Misalignment becomes axions one with attractive option [ assuming that the axion abundance after dilution is sub-dominant. A misalignment angle Given these crucial effects of the saxion condensate, what are the leading candidates and very low reheat temperature of(10 the saxion for dark matter: and the axion multiplet doeshas not couple a to large the decay Higgsan branching superfields. ratio axion Thus to contribution the saxion axions totheories. typically and dark the radiation decay that of is the excluded saxion [ condensate gives V cosmological problem. Even if weis choose still the IR contributionbelow from the superpartner freeze-in, mass which scale. cannotMD Hence be era we suppressed consider that unless larger has we important select consequences for dark matter abundance. the energy density of thethe universe, saxion producing condensate an early decays, matter-dominatedand large (MD) axion entropy era. contributions is to When created darkonly that matter. for has The the a conventional key restricted picturedominates, effect of case allowing dark on the of matter both usual survives favorite LSP cases of LSP freeze-out or axion misalignment with V JHEP03(2017)005 , . 7 PQ V ]. as high 36 and , where we PQ R V 3 T , and perform ˜ G producing axinos R and T 3) allows a . (0 O forest observations while at . α i θ ]. In other work a saxion con- ]. An alternative solution to the , by identifying the characteristic 31 2 33 , ], at the LHC and future colliders. 32 28 decays to superpartners the late decay GeV [ s 15 , and UV scattering at ˜ a , we discuss key consequences of our scheme GeV, whereas 4 → 14 – 5 – χ 10 ] and axinos [ ]. We avoid this by making the axino and gravitino . 27 29 than considered in this paper, is given in ref. [ PQ ], and further work considered mixed axion/neutralino V . In section PQ 34 V PQ V , which has no effect on BBN. . We consider two possibilities for the LSP: ˜ ˜ G we present our results for two classes of SUSY spectra: high scale aa ˜ a, 6 → and ˜ g ]. A KSVZ scheme with a saxion condensate relevant for gravitino dark g 5 35 as a function of 2 ], which also considered the possibility of gravitino dark matter from inflaton / 3 30 Y GeV. (1) is sufficiently small for 16 In sections We describe the saxion condensate MD era in section There is a considerable literature on the effects of a saxion condensate on LSP abun- Another aspect of the gravitino problem is the powerful constraints from Big Bang We work in a generic framework of TeV scale supersymmetry, taking the saxion and and ∼ O ˜ a i the same time solving issuesat with cosmological colliders structures resulting at from smallas scales. the well decay Displaced as signals of the the axion LOSP dark to radiation resulting axinos from and(i.e. decay gravitinos “gravity”) of and are the low discussed, saxion scale (i.e. condensate. “gauge”) mediation, respectively. The LSP relic density production mechanisms for axino and gravitinoaccount are discussed for in the detail dilution in section forY the saxion condensate andwith give the numerical axino results and for gravitino thecomponent as yields of the dark two matter lightest that superpartners. isIt warm, The is with NLSP free intriguing decay streaming that leads length this as to result illustrated a can in be figure consistent with Lyman- with axino or neutralinodark LSP matter [ [ matter, but at lower values of temperatures associated to this epoch and giving analytical expressions for them. The in ref. [ decay. This was further developedof by a identifying hybrid PQ-breaking inflation model fieldsgravitino with as problem vevs the of involved waterfall order a fields 10 densate very was light, used keV to axino obtain a [ PQ breaking scale as large as the unification scale in theories partner (LOSP) and thethe gravitino two [ lightest superpartners.the The NLSP LOSP it then decays decays to before ˜ BBN and if thedances. gravitino The is gravitino and axino problems were solved by the decay of a saxion condensate identifying regions of bothconnect cold the dark and matter warm cosmology withproduction dark displaced and matter. vertex signals, decay arising to A from gravitinos superpartner key [ feature of ournucleosynthesis work (BBN) is arising to from decays between the lightest observable sector super- condensate is not too large.of On the the condensate other typically hand, overproducesare if LSP kinematically dark forbidden matter. and focus Henceof on we axinos two assume from mechanisms these neutralinos for and decays LSP charginos,and production: ˜ gravitinos, IR freeze-in a systematic analysis for the dark matter abundance over a very wide range of as 10 axino masses of order thegravitino TeV which scale, could as be well as muchTeV-scale lighter. all superpartner other Freeze-out superpartners gives spectrum, too except, although low perhaps, an a the abundance bino-like for LSP a is typical a possibility providing the θ JHEP03(2017)005 , s m (2.2) ∼ H GeV. For 14 − 13 10 > GeV, saxion oscilla- I 10 s ). The energy stored in 0 10 GeV & 10 R , GeV. The onset of the saxion 10 T Pl , and make lifetime predictions 18 , namely with the same behavior & 3 M 8 s 10 − R ) GeV (2.1) a m T × 10 , whereas the analogous results for low 4 as p . (10 1 4 a 11 = 2 ) ∼ O Pl and osc T 9 – 6 – M P l ( 10 ∗ M g s 2 π m p = > < R osc T T GeV, they start during inflationary reheating. . We supplement our work by appendices with useful results . A remarkable signal of our framework, which holds regardless from the minimum today. After inflation ends, Hubble friction 10 I 16 15 s 10 . and , and R T 13 14 , theories. This contribution is sub-dominant when GeV, we find that the extra thermal saxions still fail to provide the dilu- 12 + , 14 ) and the reduced Planck mass − 10 the saxion mass, when the saxion condensate oscillates around its minimum. The T 13 ( s ∗ 10 g m In addition to the saxion condensate that arises from inflationary dynamics, the sax- The specific temperature where saxion oscillations begin is crucial for the evolution of < I bath oscillation happens during an earlysaxion Radiation oscillations Dominated red-shifts era with the (RD of scale non-relativistic factor matter. Ifthe the radiation condensate energy is long-lived and enough, the it universe eventually enters takes over an early matter-dominated (MD) era at a Saxion oscillations begin when the radiation bath has a temperature where we introduce the effective number of relativistic degrees of freedom in the thermal s tion necessary for thecontribution overproduced of axinos saxions is from never UV relevant scattering. in the2.1 Therefore, parameter space this of thermal High interest. reheat temperature after inflation: in the case where ion can also bevia generated the from dimension-5 the operators thermalin generated processes the by such DFSZ the as heavy the colored scattering fermions with with gluinos PQ charges of the reheat temperature after inflation and describe the cosmologiestions separately. start In during the the case radiation-dominated era where after inflationary reheating has ended, while with energy density stored inslower such than radiation, oscillations and red-shifts eventually like dominates non-relativistic the energy matter budget. atthe a saxion condensate. rate In particular, we identify two main regimes according to the size 2 Saxion cosmology Inflationary dynamics sets theplacing initial it conditions by an for amount keeps the the saxion saxion condensate, field fixed typically until dis- the universe expands and cools sufficiently that 3 scale are in figures of the mediation scale,with is displaced the events supersymmetry at breaking colliders.in parameters figures We study in how figure employed these in lifetimes our vary analysis. for high scale mediation is shown in figures JHEP03(2017)005 ) 75, . NA (2.5) (2.6) (2.3) (2.4) . This is ]. In the s . We plot . Γ 2 (left panel) ) = 228 26 , , GeV there is ∼ M 8 2 5 T PQ , saxion decays 13 ( H V ∗ GeV 10 g NA PQ = are not defined and T V . 15 I GeV s PQ ) = 10 V NA 15 PQ osc T V , T 10 1 2 ) era. However, radiation ( . 4 ∗ A M g 2 and T s 2 5 are shown in figure µ ) m M M Rs T I T T PQ s 3 2 I ( GeV V ∗ s , the initial saxion energy density is 15 g 2 and PQ 10 µ . 30 3 π 10 V occurs at temperature TeV Rs NA = 1 2 T s T 3 1 2 µ NA , m s M – 7 – 4 T m D M osc TeV T denotes the number of final states kinematically 13 10 T to MD 1 4 D ) A ) µ ) M osc , we enter a non-adiabatic matter-dominated (MD TeV for two different initial conditions: T T Rs 75 10 TeV = 8 for the full MSSM). The non-adiabatic era ends once ( ( . T ∗ NA ∗ 2 5 ( g ' g T D 10 PQ ∗ 75 and . g V 2 I M s 4 D 2 s T µ m q 4 5 µ ) = 10 q ) for the saxion visible width, valid in the decoupling limit and for Rs T B.8 ( ∗ . This characteristic temperature is found by imposing that the saxion 10 MeV g 2 GeV . (right panel). In the first case, we notice that for M 0 = 4 for SM and ' T Pl ' D Rs . The transition from MD M T β NA = T I s The three characteristic temperatures We require that saxion decays to R-odd neutralinos and charginos are kinematically This early MD era consists of two distinct phases, as detailed in refs. [ their values as a functionand of no matter dominated era.not For enough such to low overtake values radiation of before it decays. Thus, accessible ( the saxion condensate decays, reheating the universe at the temperature large tan where we used the radiation bath has the reheat temperature forbidden, and that decays to axionsby are sub-dominant. decays to The saxion MSSM width Higgs isexpression thus bosons in dominated and eq. longitudinal ( electroweak gauge bosons. We use the dominates, at a temperature era, where the totalreheat entropy is the universe not and conserved. adecay At large when temperatures amount the below of Hubble entropy parameter isthe is released. beginning of of Finally, the the most order last of the phase the saxion of saxions decay a width, radiation-dominated universe (RD), which starts when beginning, the radiation energy densityWe is dub dominated this by initial theconstantly red-shifted part produced initial an component. from adiabatic saxion matter-dominatedfor visible most (MD decays of the red-shifts radiation slower present and in the in universe. the Once end the accounts contribution from saxion decays where we use theergy. conservation Only of for the thecorresponding purpose total to of entropy the an full to analytical MSSM properly estimate, field red-shift content, we giving set the saxion en- and the radiationsolving bath the equally equation contribute to the energy density, or in other words by temperature JHEP03(2017)005 , , 9 16 A A PQ 10 PQ V with GeV V (2.7) (2.8) (2.9) = 13 , MD 0 PQ 15 . I V 10 s Rs T era . era ≥ A 2 NA 14 i RD' era 10 T NA MD MD T ≥ , ranging from 10 Pl ≥ 13 = 4. The RD P l (GeV) i NA =M I 10 T T , whereas in the MD s D M PQ as functions of 8 V increases from 10 − = this critical value of 5 12 T ) Rs I M NA Rs . . c T T T s T 10 ( 3 PQ PQ ∝ Rs V V GeV T 3 = 1, and PQ i GeV RD era and a 11 Rs V as a PQ 15 ≥ µ a V 10 q 15 i 9 10 NA T 10 NA T 5 2 T 3 ≥ 10 3 Rs , i 2 9 6 3 T 1 T 10 12 -3 M ≥ ) s GeV. On the other hand, the )

10 10 10

T µ i . We deﬁne

i NA 10 10

m ) ( GeV T Rs T 16 = 1 TeV, T T GeV ( Rs I T , we obtain the following dilution factor s ∗ s ( T ) 2 g ∗ 15 15 5 m 4 16 Rs GeV. g – 8 – 10 T 10 2.6 = 10 i = era Rs M era T µ µ T M Rs 10 A TeV T NA s i T T 15 Rs µ RD' era S m MD GeV to 10 ' ≥ MD S 10    5 2 ) and ( 10 R ≡ ≈ 5 4 T NA D 2.4 ) ) T 14 i i µ 5 TeV T T 10 ( ( D D 1 2 era, we make use of the scaling PQ i 13 (GeV) T . With 4 =V D 10 I 3 s PQ NA 100MeV − V from eqs. ( T µ 12 5 2 M NA Rs q D − T T T µ ∝ 10 as the temperature when dark matter is produced, e.g. the freeze-in tem- increases from 10 q 3 i 6 5 i ) in the left (right) panel, a T T RD era PQ P l 11 10 10 ), deﬁned as the ratio of the entropy after saxion decays to that at an initial V i 10 M      ( T . Cosmological eras for ( as ' ) D , RD eras are individually shaded. i PQ 10 15 V T 3 9 6 1 A physically meaningful and useful quantity is the amount of entropy released by 10 -3 12 ( NA

10 10 10

10 10

) ( GeV T I The dilution factor is extremelyto large 10 for the initialcase condition has a much small dilution, varying from 1 to 10 We can thus find We interpret perature. In theand MD RD eras, saxion decays during thefactor reheating process. We quantifytemperature this by introducing the dilution MD there is no significant entropycorresponding production to at the intersection of the three lines in the left panel of figure Figure 2 s JHEP03(2017)005 ] 0 osc 24 T , (2.12) (2.10) (2.14) (2.11) (2.13) is now 23 H . . ). 1 3 1 2 . 2.2 . s 4 R M m T . Assuming that , defined as the point TeV ) ) s ). This in turn gives a c GeV R ( PQ M m . T 2 3 V T 10 GeV D ( . ' 2.13 ∗ 10 8 10 g H 2 µ 30 10 π TeV 1 4 R 0 osc T . T 1 6 s ) = R m M 2 T TeV 4 , when 3 D T ( ) 0 osc , T > T 1 4 ) R T 1 3 ρ R 0 µ 2 osc R intersect. Its value is obtained by imposing q , and this condition is satisfied as long as T ) 1 4 T T ( = 4 2 ( R ∗ 75 ∗ 0 T osc Pl 2 g . R g 3 3 R M – 9 – T T ( M GeV T T > T still red-shift as non-relativistic matter, and once ∗ ) ) Pl 228 2 2 I g / 13 R s M 1 ) M 0 0 2 s T osc osc ) T s 10 ( T ( T R 1 8 ∗ m ( ∗ m g ∗ T , gets maximally depleted by g ( ' g ) ) ∗ 1 8 ) = 1 3 g R T < T R 75 NA . R T T π 2 ! ( 2 ( T s ) 5 ∗ ( √ 2 > T P l 228 ρ s g , which is now found by imposing the condition ) s 6 √ i 0 ρ osc R M M T m T 2 T T ( ) = ( µ ∗ ∗ ) = g M g also allows us to quantify the critical value π D T 2 T GeV 8 3 ( ( π √ D s ), we obtain √ 10 H µ GeV, namely the oscillation temperature given in eq. ( 5 ρ 4 q 10

2.11 10 = = 10 = ) ' 0 c osc ( PQ T GeV. Each comoving number density frozen-in or -out before the entropy release, osc V . R 16 The saxion oscillations for During inflationary reheating, the temperature dependence of the Hubble parameter is The quantity T < T = 1 R Afterwards, we have the conventional thermalis history the described above. initial condition Thedifferent only for difference expression the for saxion energy density as in eq. ( the inflaton finally decays the energy density stored in them results in This equation makes senseT only if is less than the maximumgiven temperature by reached eq. during ( this reheating era, so that early times before the inflaton decays, gives the expression for the Hubble parameter [ Oscillations for the saxion condensate begin at However, this need not beinflation, the which case is as setlow we by do the not decay know width the of temperature the of inflaton. reheating Wedifferent after now than extend during our a analysisBoltzmann radiation equations to dominated describing era. the evolution An of approximate inflaton solution and to radiation energy the densities, coupled at 2.2 Low reheatSo temperature far after we inflation: have assumed that saxion oscillations begin during a radiation dominated era. namely at temperatures where the three lines in theD left panel of figure to 10 JHEP03(2017)005 ≥ i T . (2.17) (2.18) (2.15) (2.16) (2.19) 3 . Rs when T ). However, . R ≥ T i 4 5 2.6 0 T NA 3. We avoid this T ≥ , √ s ≥ / ρ 0 NA i GeV s , PQ P l T T V 11 R M T = Γ 10 ). This expression breaks are shown in figure 2 ≥ 4 5 T I 2 5 ) Rs s 2.12 GeV Pl PQ I T T V s 11 ( M ∗ s GeV 10 g PQ . m V 2 and s 11 TeV 1 5 1 2 30 π ρ 4 10 75 ), evolves according to s ! 0 NA Γ s 2 5 H = T 2 I 4 5 2 m − PQ TeV 2.12 s , 5 V R + 4 R 8 = s 0 2 s 2 M T ) µ T 4 s m T 2 I m I GeV 0 s osc dt s µ dT

2 s T – 10 – 16 Hρ TeV 2 ( of eq. ( 15 , which happens if m 10 R ∗ 2 + 3 0 T T osc g ) ) GeV µ s T 1 5 R ln TeV 4 0 ln osc T dt GeV R dρ d ( T d T 10 ∗ ( 10 5 2 1 3 ∗ g × g 10 4 GeV R 2 D 5 ) or at . 2 is of course unchanged, and still given by eq. ( T 30 1 + π 2 10 5 2.2 = ' ) Rs 10 . The energy density of the saxion condensate, after the onset of T i GeV T I 1 5 0 M ( s T TeV ∗ 16 T g 2.1 4 10 Rs 2 of eq. ( 5 T 30 π 5 − µ 0 becomes larger than M q osc 10 T T 5 0 632 GeV M × T 4 10 ' =      0 NA )= T i T . The three characteristic temperatures The radiation bath temperature evolves according to We begin with the case where the saxion starts to oscillate after inflationary reheating, Moreover, also the expression for the dilution factor changes The expression for ( 0 0 NA D The redshift due to thesaxion Hubble total expansion decay is width. accompanied by the term proportional to the presented in section oscillations at very useful for an orderfor precision of calculation magnitude of estimate relic ofequation densities. the system In describing effect. this the work However, evolution we theybath. of solve are the numerically saxion the not Boltzmann condensate suited coupled to the radiation We observe the remarkable factT that the dilution factor is proportional2.3 to Field equations The analytical expressions for the characteristic temperatures and the dilution factor are situation because the saxionoscillate condensate and dominates consequently the the Universe energy enters density inflation. before itthe starts temperature to for the transition to the non-adiabatic phase is changed where in the lastdown step when we have used the expression in eq. ( The solution of this equation simply gives JHEP03(2017)005 , . ) 16 0 A T T 10 ( with

era ∗ era g (2.20) (2.21) (2.23) (2.22) A NA , MD 0 PQ 15 RD' era MD

MD V 10 2 -2 1 10 10 ), and 14 10 2.19 2 )  * 13 by = 4. The RD (GeV) I GeV s 10 0 =(M 18 I D PQ T s as functions of V 10  12 Rs . R T GeV 10 T 6 0 M . 10 T 4 = 1, and and 11 T ) , , µ RD era NA M 10 Rs 3 q 0 T s T' T' T 0 ( NA ρ T . ∗ T s 4 g M 10 , 2 T 9 6 3 T 1 ) 0 10 12 -3 30 M π = Γ ) 10 10 10

T T

10 10

( ) ( GeV T + M ∗ = 1 TeV, rad T s g , whereas some of the shadings are removed for ( s ρ 2 ∗ 16 2 m 30 π GeV. g Hρ r – 11 – 10 2 10 = = 30

P l

era era are needed to specify + 4 A 10 µ I

NA M = MD 15 s rad RD' era 3 ≤ rad MD ρ 10 dt √ Mi R dρ ρ and T = R 14 ) and our numerical studies are completely insensitive to is a constant, the equation takes a more familiar form T H 10 ∗ 2.4 g 8 6 4 PQ 13 (GeV) 10 10 10 =V 10 I s PQ (GeV) V R T 12 ) is at most of few percent. In our work we use eq. ( 10 RD era 2.21 M NA Rs ) in the left (right) panel, T' T' T ∗ 11 is defined in eq. ( M 10 ( is the effective number of degrees of freedom contributing to the entropy density. . Cosmological eras for M ∗ g T , RD eras are labeled similar to figure PQ 10 V 3 9 6 1 Finally, the time-temperature relation can be found by solving the Friedmann equation 10 -3 12 NA

10 10 10

= 10 10

) ( GeV T I The initial condition for this case is set at some high temperature where degenerate at 1 TeV. This approximation is certainly validrelativistic. at At very lower high temperature temperature, theby where approximation the using breaks full down eq. and spectrum ( the is is error one computed makes using the masses of the SM particles and of the SUSY particles, assumed where the radiation energy density results in MD clarity. In the right panel, both where In the limiting case where Figure 3 s JHEP03(2017)005 . , I ρ B I PQ V (3.4) (3.5) (3.1) (3.2) (3.3) , and thus the osc . ), respectively. On /t FI C B.16 ) with the modifications ) = 2 osc . 2.18 t s ( ) and ( H /m B.14 = 2 . FI osc ˜ ˜ a , a . C t . theories) and gravitino UV production. ˜ aZ , = + → → . We also comment on gravitino freeze-in is governed by the Boltzmann equation ˜ a ∗ ) to the one describing the evolution of the ˜ ˜ ˜ ah, aW a ∓ Z M i n Hn e W – 12 – N 2.19 → → = i + 3 i h, I e i i C s e ˜ a N e e N C dt ), and the inflaton energy density on the right-hand dn ) and ( and is set at the time 2.19 2 I 2.18 PQ s in this case dominated by the inflaton energy density. Since 2 s V m H = , which takes the same form as eq. ( I I ρ ) = s . Furthermore, we need to add the inflaton decay contribution Γ . Explicit decay widths relevant for this case are given in appendix I osc ˜ a , with t Γ s ( respectively. s → m 6 ρ → ). Secondly, we set the initial conditions for the saxion oscillation at the χ s ) = and 2.22 5 osc t and Γ ( I H ρ For a light axino, lighter than all the MSSM superpartners, freeze-in comes from neu- For the case where the saxion starts to oscillate during inflationary reheating, the → s the contrary, for aproduction heavy is axino, dominated heavier by than the all inverse decays the neutralinos and charginos, freeze-in The partial widths for these channels are given in eqs. ( Our goal here is to provide the expression fortralinos the and collision charginos operators decays to axinos 3.1 Freeze-in production ofThe axinos freeze-in productioninverse of decays, axinos ˜ is controlledThe evolution by of neutralino the and axino number chargino density decays and and we consider both production, a sub-dominant source. Theindependent results of presented where here the are axinothe completely framework and general to gravitino and two sit specific in spectrain corresponding the sections to superpartner High spectrum. Scale and We Low apply Scale mediation 3 Axino and gravitinoIn production this section wecondensate quantify effects. axino We consider and threein, gravitino different axino production mechanisms: UV by axino production production accountingFor (present from for each only freeze- the in case, saxion DFSZ we show results for the comoving number density as a function of side of eq. ( time 3 inflationary reheating is ininitial an condition MD era, we can identify 3 numerical setup needs toanymore be and extended we couple asinflaton eqs. follows. energy ( density Firstly, weρ cannot ignore theto inflaton the right-hand side of eq. ( JHEP03(2017)005 , R T (3.9) (3.6) (3.7) (3.8) & (3.10) Φ M , , ] ] , /T . /T ] i i ] e C e N , where the saxion GeV and TeV scale GeV. In each of the /T m /T ˜ ]. A complication in a . m [ 4 ˜ a 12 [ 10 1 1 m 17 [ m [ 10 10 K [ 1 K 1 theory we take R 2 ∼ K & 2 inverse then the UV production is K T + − 2 R i c FI 2 R i PQ e T C C e N T ˜ V a T ˜ a m T + m m < T T m Φ with with H decay M ˜ aH i H ˜ aH + − i e 2.1 N c FI e → C → C i → i e C → e . The axino mass is expected to be of order ˜ a N + ˜ a R )Γ T )Γ )Γ ˜ a )Γ i ˜ a i e – 13 – N m e C m inverse m − − m − n FI . i − i e C − C e ˜ a N PQ ˜ a + m ]. Here we show that the limits from UV production of V ( m m m ( ( θ ( 17 θ θ ). In the intermediate case, with the axino mass within θ 2 =1 decay 4 4 i X =1 =1 2 is model-dependent. In the DFSZ − =1 i i X X i X n 3 FI B.17 2 3 3 2 2 C GeV [ 3 T 2 π UV π π T T 6 ˜ a π 2 T = Y ). 10 ˜ a = = = = FI C . ) and ( , m R decay T PQ ]; thus decay inverse − inverse V B.15 − − n FI c 16 FI − n [ FI c C FI C C C Φ is the first modified Bessel function of the second kind. The Heaviside step makes sure that only the kinematically allowed channels are accounted for. The M θ 1 K The results for the axino comoving density are shown in figure The freeze-in collision operator is a sum of the possible sources so that the UV axinothe production gravitino is mass cutoff or at larger,supersymmetry, in which case, in DFSZ axinos is greatly ameliorateddepending by on the ( decay of the saxion condensate, with resulting limits This gravitino problem is greatlyUV exacerbated production in of PQ axinos theories also withmechanisms generally heavy occurs. provides matter a since The then particularly combination ofcomputing powerful the the upper two UV UV bound contribution production to on both axino PQ production is charges that and ifcutoff gauge the at charges, heaviest Φ, matter carrying has a mass two panels, we fixcomoving the density MSSM as mass a parameters function as of in the caption,3.2 and compute the UV axino production of axinos dilution is computed using the cosmology of section function correspondent collision operators for the charginos are where The first two terms account for the processes involving the neutralinos equation by using the inversegiven reaction, in namely eqs. the axino ( the decay, neutralinos with and partial charginos decayallowed widths masses, by we kinematics. have both types of reactions, but only the ones We use the detailed-balance principle to write the collision operator in the Boltzmann JHEP03(2017)005 ) 16 & . 10 2.9 R R T T (3.11) (3.12) 15 10 = 500 GeV, GeV is due 14 s = 2 TeV = 20 TeV with , ∼ ∼ a a 14 10 m m m − 2 Freeze-In 2.2 13 13 by inverse decay GeV 10 10 = 6. The yields of 17 (GeV) ∼ to the yield from UV GeV is 2 (4) for the axino PQ 12 PQ = 1 TeV, , ≳ 10 V g 14 V DW eq Freeze-In by decay I 10 µ PQ s Y N 10 V , for some values of = 1 TeV, and neutralino inverse 11 75 . 5 ∗ 2 10 g M ˜ 228 a 10 = m 1 10 GeV R T M 2 GeV) for the left (right) panel. g 10 9 17 and others of section 10 10 3 -2 -3 -4 -5 -6 -7 -8 -9 ). In particular, the sharp kink in each , scattering leads to an axino undiluted -10 -11 -12 -13 -14 -15 -16 -17 -18 10 − 10 10 10 10 10 10 10 10 R 10 10 10 10 10 10 10 10 10 2 GeV) in the left (right) panel. Note that 2.1 T & ∼ 10 a 3.12 Y ∗ 16 × M 16 10 8 ( 6 . – 14 – DW 10 1 × N PQ ' V 15 g 10 = 3 = ∗ 3 I g ∗ 3 (3) g s 4 14 ζ M π 10 ( 8 ln 135 13 6 3 PQ g = , the axino reaches thermal equilibrium, in which case we will V 10 PQ 6 (GeV) eq − =V = 4; while PQ = I Y 12 s PQ = 2 and 20 TeV. In both panels, 2 V 10 I D V Freeze-In by decay 10 ˜ a s × m = 2 TeV = 20 TeV 2 11 ∼ ∼ a a m m 10 ∼ Freeze-In = 2, and by inverse decay + 10 | µ 10 is the strong gauge coupling. As elsewhere, we take q ). . The axino yield from neutralino decays (red) for is the zeta function and the internal degrees of freedom UV ] ˜ a 3 9 g ζ Y 12 2.17 = 2, 10 -2 -3 -4 -5 -6 -7 -8 -9 GeV. We take GeV that have saxion dilution of section The results for the axino abundance are shown in figure To include the effect of saxion dilution, we divide the axino abundance by the dilution With UV production of axinos cut off at -16 -17 -18 -10 -11 -12 -13 -14 -15 β 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 ∼ a 10 for a sufficiently low use the equilibrium valueof of the the curve yield is inscattering. due eq. to In ( the the transition left panel, from the the smooth equilibrium change value of the slope at factor computed numerically. Sincethe axino entropy, UV the production dilution factor occurssaxion is before simply MD the the era. saxion ratio injects of Theand the analytic ( total formulas entropy for before the and after dilution the factors are10 also given in eqs. ( where (gravitino). where axinos and gravitinos should not exceed the thermal equilibrium value tan yield [ Figure 4 decays (orange) for Y JHEP03(2017)005 16 ≥ 10 . In R GeV, R T (3.13) 15 T 10 because 13 11 9 7 = 2, and 10 10 10 10 10 10 R µ for (GeV) T q (other than 14 R . 6 T R 10 ). In the left R T T Axino UV Production 13 GeV 2.2 10 , 16 . In the right panel, = 1 TeV, (GeV) ! 3 12 2 µ PQ / V 2 i ) is the gaugino mass of 10 2 3 =3⨯10 . t = I and m s m (section s 3 11 2 ,A DW m 10 3) , because the condensate is too 2.1 /N 2 , 1 + ) c 10 PQ (1

( PQ V ) 10 m . In the case where 2 R ) are proportional to GeV and growing for smaller √ T 9 < V PQ ( = ( i 10 = V 10 i -2 -3 -4 -5 -6 -7 -8 -9 γ 3.11 -10 -11 -12 a PQ 10 10 10 10 10 10 10 10 10 m f 10 10 10 i V ∼ a X & Y R 16 T – 15 – 10 GeV R ). 02) and 13 11 9 7 T . 10 0 10 10 10 10 15 , 10 (GeV) 10 4 2.17 GeV) for the left (right) panel. We take the axion domain . R 12 0 T ]. The undiluted abundance of gravitinos from scattering 16 GeV for − ). 14 3 − 10 15 13 -term of the top Yukawa coupling. 10 g 10 – × A 10 25 ) and ( . × in Axino UV Production 13 13 [ 0 6 ∼ , R 2.9 10 = 3 PQ , we calculate the final gravitino abundance by dividing the yield R ) ) and production in eq. ( T c (GeV) 08 T I ∼ ( =V . PQ s I 12 0 ( s GeV) with saxion dilution of section PQ , there is no dilution for 3.2 V 2 , 2.17 V / 10 UV 10 3 PQ 02 PQ Y . V V = 6 for the axion decay constant 10 11 (0 = ] 10 = , with ≤ ) ∼ DW I SU(3)) and the c I ( s , R s 38 ) PQ N 10 , T R 10 T 37 . The axino yield from UV scattering. In both panels, ( > V i 9 γ SU(2) is [ , 10 PQ -8 -9 -2 -3 -4 -5 -6 -7 GeV ( The numerical results for the gravitino abundance are shown in figure Similar to section R -11 -12 -10 V 10 10 10 10 10 10 10 10 T = 4, while 10 10 10 10 ∼ a 10 panel, with small for the saxions tofor dominate before they decay. The key feature is the rapid dilution where (U(1) by the dilution factor computedare numerically. also The provided analytic in estimate eqs. of ( the dilution factors perature after inflation, at both dilution in eq. ( the mild dependence on 3.3 UV production ofIn gravitinos supersymmetric theories UV production of gravitinos generally limits the reheat tem- wall number to the emergence of thethe saxion dilution MD effect era is demonstrated present inone for figures interesting the feature entire is range that of the diluted abundance is (almost) independent of Figure 5 D Y JHEP03(2017)005 is 16 × → 2 10 i / (4.1) 3 e GeV, C = 3 Y 15 / i 10 13 11 9 7 I 10 s )). e N 10 10 10 10 ( 10 = 1 GeV (GeV) 14 R 3/2 < 3.13 PQ T 10 m V R . The resulting T = R 13 GeV Gravitino UV Production Q , as long as † T = 4, and the unified I 10 16 2 s / (GeV) 2 3 D ; the spectrum is not ) in eq. ( 12 a . For XQ PQ c R † /m V 10 ). We checked that the =3⨯10 1 T I PQ = 2, ( ' s i V µ θ X ∝ 11 γ Q q 2 3.13 c 2 10 d / 3 Y Z 10 10 2 Q 2 mess = 1 TeV, c ). For the same reason, gravitino UV 9 µ M 10 -2 -3 -4 -5 -6 -7 -8 -9 (other than -10 -11 -12 = 10 10 10 10 10 10 10 10 + B.29 10 10 10 R s T 3/2 m )–( aα Y W to be comparable , as shown in eq. ( 16 ), has a SUSY breaking F-component. We take α 2 a – 16 – / Q B.25 10 3 c A.9 m 15 10 θ X W and , leading to dilution at low ) 2 = 1 GeV GeV is large everywhere, so a saxion dominated MD era c a d ( c PQ 14 16 3/2 V Z 10 = 1 GeV. The yield scales as m 10 2 / × 13 Gravitino UV Production 3 a mess . Superpartner masses arise from the effective operators c 10 PQ m (GeV) M = 3 =V I , defined in eq. ( mess 12 s ∗ PQ = V 10 X M M GeV GeV GeV GeV 9 7 = 13 11 11 mess I 10 s L = 10 = 10 = 10 = 10 R R R R 10 T T T T 10 . The gravitino yield from UV scattering and saxion dilution for 9 10 -6 -7 -8 -9 -2 -3 -4 -5 ]. These freeze-in processes are IR dominated and independent of -10 -11 -12 GeV) in the left (right) panel. In both panels, 10 10 10 10 10 10 10 10 10 10 10 39 [ 16 3/2 ˜ Y where the superfield the model-dependent coefficients split. 4 Axino and gravitinoWe as classify the the superpartner lightest spectra superpartners symmetry according breaking, to the size of the mediation scale for super- of the gravitino mass. This is(goldstinos), just as a manifestly consequence shown of in the productionproduction eqs. of ( is longitudinal gravitinos also enhancedgravitino for FI contribution small is always sub-dominantaxino compared FI to in those the from parameter gravitino space UV of and interest, and therefore we do not consider it in this work. 3.4 Freeze-in productionGravitinos of are gravitinos also producedG via decays ofyield, thermal which charginos is and proportional to neutralinos, the decay width to the gravitino, is enhanced for low values below its equilibrium value. the right panel occurs at much lowerthe values final of diluted yield is nearly independent of Figure 6 10 gaugino mass is 2 TeV and JHEP03(2017)005 ]. (4.2) (4.3) 50 , 49 . . ) of 5 i B.25 ˜ a ˜ a m ]. Baryonic feedback effects m 1 TeV 1 TeV 46 2 2 15 log / . 3 ], bounding the free streaming length , which can be broadly identified with m 45 Pl – 100 GeV 1 + 0 h M 43 . Remarkably, the gravitino free streaming 2 2 / / 3 3 sec – 17 – 4 m 10 ˜ a ], although there is much debate on this [ m of order × ), giving a lifetime from eq. ( ˜ 1 TeV a 2 48 . , m 1 mess 47 B.22 ' M ˜ ]. As we will see in this section, the origin of the current dark G ), which for the lifetime above reads a 6 Mpc 42 . 1 → – forest observations [ 0 C.8 ˜ − a , where the gravitino is much lighter than other superpartners. 40 α Pl ' = Γ M ˜ G FS ˜ a λ τ 1 Mpc. Interestingly, if the free streaming length is consistent with large we consider ' 5 mess M , 6 If the axino is the NLSP, it decays to longitudinal gravitinos (i.e. goldstino) via the Before we discuss the high and low scale cases in detail, we highlight the main features For both high and low mediation scales, axinos and gravitinos are produced in the In section This expression holds aslength long depends as only on the axino mass, and for TeV scale axinos is not in conflict with effective operators given in eq. ( The associated free streamingby length the for expression free in streaming eq. gravitinos ( is approximately given to be less than scale structure and not too smallare indicated it by can simulations address of some collisionless largecan cold scale explain dark structure matter some (LSS) [ discrepancies issues that [ LSP masses and thecoupled NLSP to decay the lifetime. radiation bath,their The and momenta LSP by thus is pure DM a free particlesmological perturbations gauge streaming. coming at from singlet This small NLSP extremely has scales decays throughconstrained the weakly free just effect by streaming, lose of Lyman- and potentially this scenario washing is out severely cos- of our framework. 4.1 Warm dark matterDark matter from from NLSP LSP production decays isproduction always cold. and On decay the could contrary, dark be matter from hot, NLSP warm or cold, depending on the ratio of NLSP and Furthermore, since the finalsubject to products BBN are limits thematter [ LSP abundance is and typically thepeculiar due axion, parameter to space either these regions. NLSP decays or are LSP not production, unless we consider relative hierarchy between them, and consider both axino andearly gravitino Universe LSP through cases. theprocesses various produce mechanisms both discussed the NLSP inmatter and the abundance, the previous LSP, which and section. today theyare all is These contribute produced made to of in the present LSP NLSP dark particles decays, only. which Highly make relativistic a axions negligible contribution to dark radiation. “gravity mediation.” Thesuperpartners. gravitino for This this has casesection cannot to be contrasted much withWe lighter focus the than our low the attention mediation onthan other all scale the the spectra case other where superpartners. elaborated the However, in axino for and gravity mediation the we gravitino do are not both commit lighter to any JHEP03(2017)005 ). , is 2 (4.5) (4.4)

= 0.5 Mpc / in ap- B.18 λ FS 3 650 GeV m 3 FS . & λ 10 ˜ a

= 1 Mpc ●● FS m ˜ a λ isocontours in m FS ˜ a λ = 2 Mpc m 1 TeV FS (GeV)

λ ∼ a m 2 / . 1 3 = 4 Mpc λ FS 2 / 2 3 ) plane. / 3 2 ) can be used for both High and m ), to account for decays to axion / 2 TeV 3 m 1 TeV . 4.2 2 , we only need to require / , where we draw , m 3 B.20 6 ˜ a 7 m 2 m sec 4 3 2 10 1 log 6

10 10 10

/ 2 3 ) ( GeV m 10 × 1 + 0 5 . – 18 – 8 ' ˜ a ˜ a for the warm dark matter component. In the left (right) 3 a m 10 1 TeV → = 0.5 Mpc 1 FS

FS ˜ − G

λ λ ●● 2 / regime, where the free streaming length is approximated by ) is used only in High Scale mediation. In both equations we 1 ). The red and blue dots label the spectra we choose for High Scale = Γ 2 = 1 Mpc / 2 ˜ 4.4 aa 3

FS /

λ 2 3 / m τ (GeV) → 3 ∼ a ˜ m G m 2 TeV (

= 2 Mpc . For Low Scale mediation in section ˜ a λ FS 5 ˜ m Ga → ) plane. These results are derived by using the full derivation of a 2 2 Mpc . / 3 1 forest observations and in the correct ballpark to address LSS anomalies. Nu- . In the . Free streaming length ' , m α ), the isocontours are vertical lines in the ( ˜ a C 2 ˜ a FS m 4.3 For gravitino NLSP, the decays to axinos are mediated by the operator in eq. ( λ 2 3 10

10 10

/ 2 3 ) ( GeV m are comparable.approximately The given axino by free streaming length, in the limit where To simplify the discussionbelow the we other take superpartner the masses.Low Scale axino Thus, mediation, while mass eq. eq. to ( ( have be ignored phase of space order factors 1 TeV that — become i.e. relevant when not the far NLSP and LSP masses We assume that the decaythus to the decay saxion width and is axinoand half axino ﬁnal of only. state the The is expression resulting kinematically in lifetime forbidden, eq. is ( and merical results are shownthe in ( the left panelpendix of ﬁgure eq. ( decay process ˜ mediation in section since the free streaming length becomes independent of Lyman- Figure 7 panel we show the result for gravitino (axino) LSP produced through the axino (gravitino) NLSP JHEP03(2017)005 3 3⨯10 (4.6) -like , its de- ˜ 14 . Curves

GeV 1 1 e

= 10 . N LOSP VPQ M 2 2 -likeH ) as a function ' ˜ B ˜ G region, the free 2 . On the other a, 2 β M

GeV GeV / GeV

13 12 3 = ˜ 3 = 1 MeV PQ = 10 MeV = 100 keV i V m 10 3/2 12 ( 3/2 (GeV) 3/2 the neutralino mixing = 10 = 10 m are labelled by values of

m 1 m i 10 ˜ a ij M μ= 2 TeV

PQ PQ → 1 1

V V cτ R ˜ a ˜ = 2, and − N 3 m Γ β c +Z/h +γ +Z/h = ˜ ˜ ˜ , with i GeV →a →G →G 2 µ . | cτ 1 1 1 i 3 ˜ ˜ ˜ N N N 41 = 2, tan cτ . In the . Curves for 10 R µ q 7 β = const, consistently with the approxi- PQ c 6 5 4 3 2 1 300 | -1 V 10 ˜ 2 a ! 10 10 10 10 10

1 i ) ( τ m c m e 2 N 2 )). In the pure Higgsino LOSP limit, with µ | 1 MeV, the decay channel of neutralinos to / m 3 3 31 . 2 and is thus independent of

m R – 19 – B.14 / β 2 2 3⨯10 bounds are evaded for suﬃciently heavy gravitino s / 3 -like µ α 2 ˜ ≡ | q m ˜ a LOSP ) and ( k , ˜ a 1 k -likeB B becomes 1 ˜ H B.10 ˜ a 5 m k

= 1 MeV ). Lyman- = 10 MeV = 100 keV . 2

3/2 3 3/2 m 3/2 m m 4.5 10 /2= 2 TeV ) in the left (right) panel, with ' 2 µ ˜ a μ(GeV) ( =M 1 → 1 1 c ], which in our case is made irrelevant by the saxion condensate dilution. ˜ ) of appendix M M N

GeV GeV GeV Γ 52 +Z/h +γ

14 13 12 +Z/h , ˜ ˜ ˜ B.14 ≡ 51 →a →G →G ˜ = 10 = 10 = 10 a 1 1 1 cτ

PQ PQ PQ will be suppressed when the LOSP is bino- or wino-like. ˜ ˜ ˜ N N N

V V V . The neutralino LOSP inverse partial decay width ) for ﬁxed ˜ a 1 are shown in red and are labelled by values of k ˜ and are shown in purple when the LOSP is Higgsino-like and in orange when it is bino-like. M G 2 6 5 4 3 1 300 For Low Scale mediation with A related noteworthy example is the case of gravitinos coming from LOSP freeze-out ( -1 10 2 cτ 10 10 10 10 10

i 3 ) ( τ m c matrix (for details seedecoupled eqs. bino ( and wino, hand, gravitinos becomes sufficiently enhanced to dominate over the axino final state, giving a given in eq. ( Here, we define the mixing factor 4.2 Displaced signals atIf colliders the lightest observable-sector supersymmetriccay partner to (LOSP) the is axino a and neutralino vertex. Higgs/longitudinal Z The bosons will “lifetime” leave for missing this energy and channel a can displaced be obtained from the associated decay width mate expression in eq.and/or ( axino, and issues with LSS can be addressedand again decays [ by TeV scale superpartners. for m The LOSP lifetime is given approximately by the smallest The full result isstreaming shown isocontours in are the along right the panel lines of figure Figure 8 of JHEP03(2017)005 ] )– 54 (4.8) (4.7) B.26 of order becomes µ cτ = 14 TeV this ], with a susy s . ). The change of ] implies a reach √ 4 µ 56 ( , 54 1 s , 55 µ M m 2 does not vanish due to is given approximately by κ 2 2 in red, labelled by values of 1 / 5) TeV, at ˜ 3 ˜ . N G µ κ 2 ) if q m ), following from the constraint cτ , cτ ] implies a reach in ˜ 5 100 keV G . 54 B.2 ) drastically change as a, 5 ) while fixing 1 4.7 ! = ˜ = (1 i 8) TeV, figure 5 of [ M 1 75 ( , e m 6. The proposed CMB Stage-IV experi- . N . = 2 TeV) in the left (right) panel reflects ,( i µ 0 ) = (1 TeV, 1 TeV, 2). Here we illustrate m 10 1 (1 MeV) 1 TeV β cτ ] ∗ < = (3 M

g 26 ]. For ˜ ˜ eff tan m G ) and eq. ( 1 , – 20 – k 53 N 1 [ 03. 4.6 . 4 D m . The LOSP lifetime 2 m 2 , we display curves for µ, M / ] is ∆ = 0 = 2 TeV ( 3 ' 8 = 14 TeV, figure 4 of [ µ 57 43 eff m ˜ 10) fb for ˜ 112 G s , N 3 → √ ' c 1 5) TeV. At a future 100 TeV collider [ ˜ . N a ν at 2 ρ Γ ρ , . The neutralino LOSP decay can lead to observable displaced 1 5 1 . In figure . ≡ − 1 are in purple when the LOSP is Higgsino-like and in orange when it as the parameter space changes, always keeping the unified gaugino M )m. ˜ we show predictions for G ˜ a M = 3 5 i 6 2 cτ cτ theory cτ = (1 10 eff , ' 0 7 of order (100, 10)m. Recently, a surface detector called MATHUSLA [ N 2 and m ∆ . In the left (right) panel, we vary theory, the axino is dominantly produced by IR freeze-in, as discussed in 5 M cτ i 0 cτ . Since the axino is lighter than the superpartners in the thermal bath, FI contains the analogous mixing factors given in the decay widths eqs. ( 11, in theories with High and Low Scale mediation for the particular point in . ] can be sensitive to ∆ )m for ˜ ˜ G 3 3.1 k 58 of order (10 ). 10 = 0 In sections The lifetime that can be probed at the LHC and future colliders depends on the , . The curves for 5 2 cτ h PQ B.29 5 Results for high5.1 scale or “gravity” The mediation DFSZ In the DFSZ section The Planck experimental bound [ ment [ 4.3 Axion darkSaxions radiation can decay tosymmetry. axions Using with the branching apredict ratio rate the of given amount the of by saxion dark to eq. radiation the ( to visible be sector [ and to axions, we the smallest behavior in the lifetime curvesthe at fact that the mixinglarger factors or in smaller eq. than ( signals at the LHC and future colliders over a remarkably wide range of parameter space. supersymmtric parameter space ofthe ( variation in the mass relation, V is bino-like, labelled by values of At the LHC, with(10 30 ab production cross section ofin (10 Ω total production cross sectiondegenerate of squark supersymmetric and particles, gluino which masses,cross we section quote ˜ is for of the order caseCMS (100,1) to of fb, reach so that plannedhas runs been of proposed the to LHC search for will (ultra) allow long-lived ATLAS particles and at the LHC and future colliders. where ( “lifetime” JHEP03(2017)005 . 16 16 9 10 10 given by )(GeV) GeV) 15 15 ∼ a GeV CDM WDM )( 10 10 ), which is ∼ a 15 ,m 2 PQ (1000, 800) (400, 800) 3/2 ,m (1000, 800) (400, 800) /V = 10 3.13 3/2 (m I GeV ). s 14 14 (m 15 ) ) 10 10 ) plane in figure * * R 2.2 = 10 (GeV) (GeV) I GeV GeV ,T s =(M =(M I I PQ PQ 16 16 s s 13 13 V V PQ 10 10 V (section =3⨯10 =3⨯10 I 0 GeV I s s 18 2.1 12 12 = 600 GeV. The top (bottom) row s 10 10 = 10 CDM WDM I m s 11 in the ( . 11 11 6 5 4 9 8 7 = 0 10 10 10 16 15 14 13 12 11 10

10 10 10 10 10 10

10 10 10 10 10 10 10 10

R R ) ( ) ( GeV T GeV T h = 1 TeV and 16 16 – 21 – 10 10 µ = 1 CDM WDM GeV discussed in section 15 15 M 10 10 10 11 from axino freeze-in and gravitino UV production. We ﬁx 2 = )10 . / . 2 ( 14 14 High Scale mediation: DFSZ M = 0 & 10 10 2 PQ PQ R h (GeV) (GeV) T and draw contours of Ω =V =V I I s s PQ PQ R )(GeV) )(GeV) 13 13 ∼ ∼ V V a a T CDM WDM = 2, and 10 10 . Both production sources are heavily diluted by the decay of the saxion ,m ,m ). On the other hand, gravitinos are populated by the UV scattering of (1000, 800) (400, 800) (1000, 800) (400, 800) β R 3/2 3/2 and T 3.9 (m (m 12 12 PQ 10 10 V = 4, tan . Contours of Ω ) and ( D 11 11 3.7 4 7 6 5 9 8 We compute the total DM abundance from these LSP and NLSP production sources 10 10 10 10 13 12 11 15 14 16

10 10 10 10 10 10

= 2,

10 10 10 10 10 10 10 10

R R ) ( ) ( GeV T GeV T µ proportional to condensate, which also makes the LOSP freeze-out and decay contributionin negligible. terms of After decaying to the LSP, the NLSP number density is transferred to that of the LSP, is for the cosmology with occurs via decays ofeqs. neutralinos ( and charginos withquarks, rates proportional gluons, to and 1 their superpartners with an abundance given by eq. ( Figure 9 q JHEP03(2017)005 16 10 GeV to 15 CDM WDM and give = 2, and GeV and 10 15 , predicted ∗ β 10 M , the dilution 10 h/Z 14 = 10 + ≥ NA I a ) T s in the right panel. * R GeV a new regime T = 4, tan 13 16 (GeV) PQ 10 =(M R I D V GeV, the blue and red s 10 T . 18 > 12 . GeV I GeV 10 = 2, 2.2 16 s 15 )(GeV) , which is much larger in the LSP is increased from 10 µ ∼ a q I m 11 ,m PQ = 10 s (1000, 800) (400, 800) I =3⨯10 V 10 3/2 s I 2 I s , while the bottom two panels have (m s GeV and 10 era and dilution becomes independent 10 2.1 + Neutralino LOSP two contours are shown. The blue one 3 2 6 5 4 1 16 10 -1 ). . We ﬁx 10 0 10 10 10 10 10 I

9 s

1 N

) ( τ m c ˜ in the left panel and 2.1 . However, for × 3 PQ PQ 10 V V – 22 – = 3 10 I (section s ) GeV. The top two panels have 9 CDM WDM 18 10 2.2 GeV 10 , and in the right two panels we take 11 using ﬁgure = 600 GeV. The left (right) panel is for the cosmology with . 15 , s PQ 8 16 = 10 m V = 0 10 I , while the parts of the contour with positive constant slope 10 s 2 GeV R = ) × * h T 16 . Hence, for I 7 3 I s , s (GeV) 10 =(M R I 15 s T =3⨯10 I from Ω s 6 10 = 1 TeV and = (10 PQ High Scale mediation: DFSZ V ∗ µ GeV )(GeV) ∼ a 18 5 GeV discussed in section where UV gravitino production dominates, the blue contours are well above M = ,m 10 10 1 R GeV and are described by the cosmology in section (1000, 800) (400, 800) . The lifetime of the neutralino LOSP, which decays dominantly to ˜ . This also explains why in the right panel, as = 10 3/2 T I GeV, the contours move to lower M s 10 )10 (m PQ and therefore of 4 16 & V 10 ( 3 2 5 4 6 1 10 -1 When UV gravitino production dominates, the diﬀerence in the slopes of the contours In the upper two panels, for each value of 10 M 2 = 10 10 10 10 10 10 ≤

N 2 ) ( τ m

c × R ˜ R of vertical contours differ slightly only because of a different results from the dilution factor,At scaling large as right panel than in thein left panel. the This right means panellower that than the production the needs left, to3 be and much hence larger theis contours entered in where the freeze-in occurs right during panel the are MD at much is an example ofvertical parts an of axino the LSP,hence contours while are have the independent axino red of FIhave one as UV is the gravitino dominant an production productionfactor example dominate. from mechanism of When and the gravitino freeze-in saxion LSP. occurs condensate The above is proportional to left two panels, we take contours for are therefore described by theT cosmology in section by determining M T and therefore we only need the LSP mass to compute the final DM abundance. In the Figure 10 JHEP03(2017)005 . , 9 9 9 and from 2 ) GeV. / R so that 3 16 T For large 4 m is so large somewhat now arises 10 . R − R PQ 1 T T V contours differ 15 where axino FI 2 2, which is often is sufficiently large. h R , where the saxion √ ), and the axino FI GeV and misalign- T / 1 DW 15 N 2.15 DW 10 . A reduction by a few by several orders of mag- N R . T a is of order (10 GeV, freeze-in occurs above by PQ f V a 16 f PQ is compensated by an increase in 10 V = 100 GeV) in figure ) GeV, are excluded by isocurvature den- R × 2 T 16 / 3 10 , as seen in eq. ( m − R = 100 GeV) in figure T 15 2 / 3 and the blue contours have larger , so that the red and blue Ω is larger than 2 m 2 GeV and 3 / / 3 – 23 – 2 3 to be sufficiently large for leptogenesis. PQ 15 m V R /m corresponding to the unshaded region of figure , of order (10 T 1 = 10 and PQ V PQ ∝ I ˜ a s 2 m / ]. Since and 3 could be much below the energy scale of inflation, lowering < V is dependent on the model for the PQ phase transition and Y R R 26 T R R , T T = 100 GeV — that misalignment axions overclose the universe T . In particular, a decrease in 19 2 R / The saxion condensate increases the maximum value of T 3 to compensate. On the other hand, at lower This allows very high reheat temperatures after inflation, . is sufficiently small, typically 0.1 - 0.3 suffices, and/or m R T LSP mis θ GeV. , so that for these theories the lightly shaded region also becomes physical. m is independent of 16 PQ ˜ a V 10 GeV for Y 14 , the saxion condensate lowers the minimal value of An important part of the axino problem is that the minimal value of with the contour for High Scale mediation ( ∗ 10 9 M × , so that the dilution factor is proportional to In the lower two panels, for Another key point emerges from comparing the contours in the two upper panels of A key point emerges from comparing the contours in the two upper panels of figure Each contour is divided into thick and thin parts. The thick parts indicate cold dark GeV to The largest possible values of = 4 8 0 NA I abundance depends on sity perturbations if there iswe do a not sufficient show contribution any ofmisalignment such misalignment excluded angle axions region, to as dark the matter. bound from In such our perturbations figures can be avoided if the axion typically give sub-dominant contributions toment dark angles matter order for unity [ an order of magnitude,although axions they are could give typically a sub-dominant comparable contribution on when all the contoursT of figure s nitude. — 4 unless the misalignment angle isdensate because very the small. decay This of the difficulty condensate is also removed dilutes with misalignment a axions, saxion so that con- they inflaton decay rate is slow, the shaded bands andorders reducing of the magnitude would maximal still allowed allow value of figure on the modelcondition for to reheating typically afterHowever, be inflation. certain theories For mayhigher instantaneous have than PQ reheating breaking we beforeWe expect expect inflation the the dark even shaded for region to be unphysical in all models. On the other hand, if the condensate is absent. 10 baryogenesis may occur atbecause very inflation only high gives temperatures. aThe saxion The condensate precise if constraint upper PQ on bound symmetry is on broken before inflation. dark matter (WDM)constitutes case more where than the 50%understanding component of possible of the difficulties dark totalmissing with matter abundance. satellite pure from problems. CDM, The the as thin NLSP in lines decay core-cusp, may be toowith the relevant big contour for to for High fail Scale and mediation ( hence need larger dominates, only because of matter (CDM) where NLSP production is sub-dominant. The thin parts label the warm the red ones; this is because JHEP03(2017)005 , . , I + + . < s PQ PQ PQ 10 /D R V V R GeV GeV V LOSP GeV, T 6 : once cτ 18 LSP I 16 < T s theories, drops FI indicated m 10 GeV have ˜ a 0 GeV, and R = 10 Y 16 > for DFSZ PQ T I 10 GeV I ∝ V s 10 PQ 6 s 2 10 V , for axino FI a 3.2 × h 9 & 3 less (greater) than . For very high , R R 15 T NA T era and the abundance . For 10 is obtained by inverting less than about 10 axis to the lifetime using ), from the values of = 10 R 5 10 NA PQ ) for ( T 4.6 PQ I V R GeV and any s V ) is required for a given I 9 GeV is vertical, corresponding ∝ s ,T , which in turn gives a shorter 10 GeV. The key feature in DFSZ 18 R (1 . PQ T × 10 9 V PQ . The large values of 3 1), with the constant value applying V , = 10 2 I > ), can be predicted by determining PQ I s 2 PQ V s R m. 10m. In the left panel, as 4.6 ∝ T /V = 7) ). As explained for figure R − D ' I T s (5 ( , so that the contours are vertical. At very low 2.1 10 – 24 – from axino freeze-in dark matter. and converting the ∝ R , we show the results for the total abundance in LOSP ˜ T 9 ∼ a 9 are steeper with Y PQ . Gravitino production is everywhere sub-dominant. cτ cτ V ) leads to a lower PQ I PQ s V less (greater) than 10 (section V , so that the dilution factor also drops and , more dilution (i.e. higher R = 0 A T 2.2 of order, or larger than, the scale of supersymmetric grand and I lead to s GeV, the contour for ]. Consequently, for I R are linear in 9 8 s GeV, UV axino production dominates dark matter production T [ 11 6 I D 10 s , the thin parts of contours give warm dark matter from NLSP decay. 10 9 theory > > and R + ˜ a T R Y , we predict the neutralino LOSP lifetime, eq. ( T 12 , and those with shows that for as a function of 3 region they are identical to the curves in figure ). The left (right) panel corresponds to the cosmology of PQ 12 V 11 R at a rate that depends on the cosmological era during FI. However, for 4.6 GeV, both GeV). For UV axino production, GeV discussed in section In figure The parametric dependence of the contours can be understood from Ω We do not show the LOSP lifetime for If the LOSP is a neutralino, its lifetime, eq. ( T ∝ the contours become sloped as the axino yield becomes dominated by freeze-in; in this is sufficiently high, the axino freeze-in occurs during the MD 10 10 10 R R PQ I fixed by the observed darkwith matter the abundance, left in (right) a panelis way for completely that, analogous for to all figure and hence, compared to DFSZ Figure to an equilibrium axinoT abundance, while the contours10 with T low where, for UV production, the10 dilution factor only if the equilibriumdashed abundance line. is reached, For which happens above and to the left of the figure In fact UV axino productionwhere dominates axino everywhere, freeze-in except becomes for important.and The gravitino blue and LSP, red nearly curves,factor corresponding coincide; of to two axino they between differ the curves. only because the LSP mass differs by a 5.2 The DFSZ In addition to thethere axino is FI also andtheories. axino gravitino production UV In a contributions from setting existing UV similar in scattering to DFSZ discussed figure in section there is atransitions very to robust occurring in prediction MD increase. of As in figure by the left panels of figure eq. ( 10 larger saxion condensate (higher LOSP lifetime. However, thiss behavior does not continuebecomes for insensitive an to arbitrarily high much of the contourthere is is still a very vertical robust as prediction freeze-in for occurs duringfrom the MD dark matter abundance.the The prediction two axes shown in in figure the right panels of figure V JHEP03(2017)005 16 16 ). 10 10 GeV 2.2 15 GeV CDM WDM 15 15 16 10 10 = 10 GeV I , the lifetime s 15 I (section s =3⨯10 I = 10 14 14 s I GeV s ) ) 2.1 10 10 * * and 16 = 600 GeV. The top R (GeV) (GeV) s T =(M =(M I I m PQ PQ s s 13 13 =3⨯10 V V GeV I s 10 10 GeV 18 18 ) ) + = 10 ∼ ∼ a a I = 10 s 12 12 I ,m ,m s 10 10 3/2 3/2 CDM WDM (GeV) (GeV) (1000, 800) (400, 800) (1000, 800) (400, 800) = 1 TeV and (m (m µ 11 11 6 5 4 9 8 7 = 10 10 10 16 15 14 13 12 11 10 GeV discussed in section 10 10 10 10 10 10

10 10 10 10 10 10 10 10

10 R R ) ( ) ( GeV T GeV T M )10 16 16 . ( – 25 – drops below the masses of the other superpart- 2 = 10 10 / & 2 2 / 3 R M T m 15 15 10 10 11 from axino freeze-in, axino UV, and gravitino UV production. . 14 14 = 2, and )(GeV) )(GeV) High Scale mediation: DFSZ = 0 ∼ ∼ a a 10 10 β 2 PQ PQ ,m ,m h (GeV) (GeV) (400, 800) (1000, 800) (1000, 800) (400, 800) =V =V 3/2 3/2 I I s s PQ PQ (m (m 13 13 V V 10 10 = 4, tan . D , unless the axino abundance from UV scattering reaches equilibrium, when 2 LSP 12 12 3 / 10 10 m 2 LSP CDM WDM CDM WDM . Contours of Ω = 2, m µ q 11 11 4 7 6 5 9 8 10 10 10 10 13 12 11 15 14 16

10 10 10 10 10 10

10 10 10 10 10 10 10 10

R R ) ( ) ( GeV T GeV T 6 Results for low scaleAs or the “gauge” mediation mediation scaleners is and reduced, LSP dark matter is composed of gravitinos. Furthermore, TeV scale axinos and (bottom) row is for the cosmology with unification, displaced vertices should be observable atscales LHC. as For fixed it scales as Figure 11 We fix JHEP03(2017)005 16 = PQ 10 1 V GeV M sec as 10 = 2 15 )(GeV) I ∼ a GeV )10 10 2 = s , predicted / & 18 ,m 2 ( (1000, 800) (400, 800) GeV) 3/2 M h/Z 14 . = 10 / (m I 2 10 + s R / T 3 a ) * m 13 greater (lower) than (GeV) 10 =(M R I R = 2, and s T T β GeV 12 16 10 GeV, so that in this case the 11 16 = 4, tan =3⨯10 CDM WDM I − 10 s D 15 11. Following the same setup as in . 10 + Neutralino LOSP 3 2 6 5 4 1 = 2, 10 -1 10 10 10 10 10 10 +

= 0

N q ) ( τ m c ˜ 2 ). The left (right) panels assume h 1 GeV decays to gravitinos occur before BBN 2 of order 10 10 / 2.2 – 26 – 3 10 . . Here we are concerned with Low Scale mediation PQ 2 . We fix / V 3 9 5.1 GeV 11 10 m GeV 1 GeV, a non-NLSP axino has a sizable branching ratio 16 theories is dominantly from axino freeze-in and Gravitino 18 (section > 0 ). 8 2 = 10 / I =3⨯10 2.1 10 3 s I 2.1 s CDM WDM m ) * 7 (GeV) 10 =(M , so that for R I s T (section theory 6 0 B.4 2.2 = 600 GeV. The left (right) panel is for the cosmology with 10 11 contours of figure , the thick (thin) parts of the curves indicate that dark matter arises . GeV). It is important to remember that the dilution factor is larger for , we show contours of Ω )(GeV) s High Scale mediation: DFSZ ∼ a 16 m 11 = 0 ,m 13 5 (1000, 800) (400, 800) 10 2 3/2 10 . h . The lifetime of the neutralino LOSP, which decays dominantly to ˜ (m × and PQ 4 9 V , even for the highest values of 3 2 5 4 6 1 = 3 10 -1 GeV discussed in section In figure 10 10 10 10 10 10

∗ ˜ hh

1 10

N = 1 TeV and ) ( τ m c ˜ M dominantly from LSP (NLSP) production.after The inﬂation. gray The region is upper excluded10 (lower) because panels PQ are breaks for( the cosmology of larger of supersymmetry breaking, as inLSP. gauge Axinos mediation, produced and from hence freeze-inof take subsequently gravitino the decay dark gravitino and matter. to produce be a the warm component ﬁgures axinos must be the NLSP. 6.1 The DFSZ Dark matter production in DFSZ UV production as explained in section given in appendix and this part ofapply the even gravitino if problem thegravitino disappears. axino dark matter. is For For not suchto light the gravitinos, NLSP, increasing our the results generality of the predictions for from the Ω µ discussed in section neutralinos decay to gravitinos with an inverse partial width of order ( Figure 12 JHEP03(2017)005 16 16 . 10 10 (6.1) ). With CDM WDM CDM WDM GeV 15 15 . For low 2 10 10 / 13 PQ 3.12 3 V 10 m

× = 1 GeV 10 14 14

2 / 2 3 GeV GeV

10 10 ). m 16 16 2 (GeV) (GeV) 2.2 PQ PQ

13 13 V V

= GeV 1 =3⨯10 =3⨯10 given in eq. (

I I

10 10

/ 2 3 s s m R GeV

/ 2 3 = 10MeV 100 m I 0 (section = 10 MeV 16 s 12 12 = 1 MeV 3/2 10

m 2.1 and 10 10 3/2 = 100 keV

m = 600 GeV. The top (bottom) row is × 3/2 2 s m / 3 3 m m 11 11 6 5 4 9 8 7 10 10 10 16 15 14 13 12 11 10 s 10 10 10 10 10 10

) is much enhanced for low

10 10 10 10 10 10 10 10

µ R R ) ( ) ( GeV T GeV T m 3.13 16 16 – 27 – 10 10 µ = 1 TeV, and

TeV =

1 GeV 10 µ

/ 2

3 = 10 MeV m m3/2 = 15 15 1 2

GeV discussed in section

1 / 2 3 , gravitino UV production is so eﬃcient that gravitinos = 1 MeV 10 m 10 10 = 10 keV R 11 from axino freeze-in and gravitino UV production. We ﬁx 10 M

3/2 . m 4 D T

m 10 GeV )10 3/2 = = 0 14 14 Low Scale mediation: DFSZ 2 = .

( / 2 3 2 = 1 keV 10 m 10 10 µ 2 2 h q PQ PQ & M (GeV) (GeV) =V =V R I I s s PQ PQ T 13 13 2 V V / = 2, 10 10 3 ), the ﬁnal abundance reads β GeV m 2.9 and high enough 12 12 12 . 2 10 10 / 0 CDM WDM CDM WDM . Contours of Ω 3 = 4, tan ' m D 2 11 11 h 4 7 6 5 9 8 2 Gravitino UV production given by eq. ( 10 10 10 10 13 12 11 15 14 16 /

10 10 10 10 10 10

= 2,

eq 3

10 10 10 10 10 10 10 10

R R ) ( ) ( GeV T GeV T Ω µ dilution in eq. ( for the cosmology with enough thermalize, with an undiluted yield independent of Figure 13 q JHEP03(2017)005 ) 16 ). − 10 3.7 = 4, (6.2) 2.1 D 15 10 = 100 MeV (red) or to 3/2 = 2, m 14 (section µ h/Z , q 10 2 as well, and with ) with the values

+ GeV

2.2

1 GeV 10 a

R between 650 GeV

16 / 2

3 / 4.6 13 m T ˜ eq 3 a (GeV) 10 Ω R m

=3⨯10

= 1GeV 1 × . We ﬁx

/ 2

s 3 12 m # 2 13 10 2.1 ) R 11 T WDM CDM 10 ( ˜ g 2 TeV (section m , corresponding to the vertical thick 10 R + Neutralino LOSP 3 2 6 5 4 1 2.2 10 -1 T 10 10 10 10 10 10

0 = 600 GeV. The left (right) panel is for the

1 GeV discussed in section

) ( τ m c s ˜ 10 m GeV R 10 T 11 contours of ﬁgure . 13 – 28 – 10 10 . Therefore, the curves for equilibrium gravitino = 0 R 9 2 2 T 10 h +γ/Z/h GeV and the dilution factor in the cosmology described +Z/h ), corresponding to the sloped parts of the thick curves 2 ˜ ˜ / 10 = 1 TeV and 3 →a →G 8 3.13 1 1 10 , the result generically applies for any µ GeV m 10 ˜ ˜ N N ˜ a < GeV discussed in section = GeV m " R 10 1 16 7 T less (greater) than 10 (GeV) M 10 ' )10 R R 2 T & T =3⨯10 h ( I , as in eq. ( 6 2 = s eq . / 2 R 10 = 1 MeV − / 2 = 10 MeV = 100 keV decreases, the gravitino may become non-equilibrium with an abundance T Low Scale mediation: DFSZ n 3 R , the LOSP lifetimes can again be predicted from eq. ( 3/2 M T 3/2 Ω 3/2 R scales linearly with m m m 5 T 14 10 WDM CDM . The lifetime of the neutralino LOSP, decaying dominantly to ˜ 2.2 (pink), predicted from the Ω ) is independent of that give the observed dark matter abundance. The left (right) panel corresponds 4 = 2, and 3 2 5 4 6 3.9 1 10 -1 γ/Z In ﬁgure In the lower panels, 10 PQ 10 10 10 10 10 β

1 + N

) ( τ m c ˜ ˜ parts. Lastly, the axino FI production dominates at theof vertical thin parts ofto the the curves. cosmology of It is worth noting that we are choosing different values of the gravitino mass in these kinematics respectively. in section production (with a constant undilutedequilibrium yield) gravitino UV are production now is atdilution, close a to the slope. being final On proportional the yield to other becomes hand, independent non- of which is valid when thethin quantity vertical inside curves label the the squareand axino ( brackets FI is domination. less Since than1 TeV axino unity. as FI Lastly, required production the by in the eqs. free-streaming ( length for cold dark matter and neutralino decay duction. As decreasing with in the upper panels. The diluted abundance is given by G tan cosmology with In the upper panels, the vertical thick lines correspond to this equilibrium gravitino pro- Figure 14 JHEP03(2017)005 2 . / 3 are R . (6.3) m 2 T / 3 and the 15 (1) MeV is shown m . Similar R O I that axino T s for dilution. 16 11 I , s 2 (purple), with the . The thin (thick) 11. Since the thick . ˜ g ˜ G 15 m , the thin parts of the = 0 2 TeV → 2 h 1 13 e 2 N dominates and the lifetime 2 ˜ a , the result generically applies / . This decay channel, with the ˜ a 3 is needed. It is remarkable that, 11 in figure → m GeV m . I s 1 e 2 N = 0 γ/h/Z (red) and ). In comparing the non-equilibrium 2 ˜ a h as contours of Ω 3.12 ). In the right panel, the purple curves are GeV → that can lead to LOSP displaced signals. In 1 15 I 13 PQ where the curves of the upper right panel of s e 2.1 N V 10 R – 29 – T × and 4 2 that gives Ω / 3 ) are independent of (section 2 GeV. Thus, the features of these parts are identical to m PQ 6 V − is sufficiently large, 2.2 3.11 5 . The prediction of the total decay rate, when dominated 2 based on eq. ( 1 TeV as required by the free-streaming length for cold dark 2 and is insensitive to dark matter production and dilution. 2 6 GeV, the gravitino UV abundance decreases with DW / / / 10 − 3 3 2 N 15 / & m 3 − m far above the TeV scale. As in figure = 1 14 R m 9 R . T enhances the production and thus requires a larger eq ˜ T G 10 0 2 ), dominates in the purple portions of the curves and gives the very ) and eq. ( theory is sufficiently low for the gravitino to be thermalized, any lower / . /Y ' 3 2 4.7 + 3.9 / eq ˜ a m 3 . To see gravitino UV domination, we first notice from figure PQ Y UV UV ˜ ˜ a G V m Y Y 13 , the relevant decay modes are between 650 GeV ) and ( , depends on ˜ a ˜ G 14 3.7 m → 1 GeV and e N 6 Finally, we also make predictions of neutralino LOSP lifetimes in figure In the lower panels, since the dilution factor becomes unity in regions where cosmology studied in section − 5 R On the other hand,prediction is once set bylines the label value of warm (cold)T dark matter. The lefttruncated (right) on panel the corresponds right to at the the low (high) values of matter and neutralino decay kinematics, respectively. to figure latter dominating for low by Γ one finds that this ratio is less than unity10 in all relevant parametercurve space becomes of thin and figure vertical, correspondingin to eqs. axino FI. ( As axino FIfor and any UV production axino UV production to that of the gravitino parts of the curves indicatenever gravitino dominates UV domination, when it canthose be in seen figure that axino production UV production generally dominates thatthermalize of axino and FI. Above black dashed lines, UV axinos due to the interplay betweenalways the within two the decay reach channels, of the current LOSP and lifetimes future for6.2 colliders. low The DFSZ The dark matter abundance is shown in figure lines are dominated byof the the axino late FI axino contribution decaybut and to there exists lead axions a and to set gravitinos. warmparticular, of dark For correlated a clarity, matter low only because oneHowever, value once of results in a decrease in its energy density and a smaller particular, we open up a— new decay the channel neutralino when LOSP the decaysrate gravitino mass to given is the in less gravitino eq. than and well ( known displaced vertex signalfor of this gauge mediation. scenario with Our framework yields a cosmology two panels to demonstrate interesting LOSP decay signals for the relevant regions. In JHEP03(2017)005 16 16 10 10 = 2,

/ 2 3 = 10MeV 100 m =1 MeV q m3/2 = 100 MeV 3/2 15 15 m

10 10

= GeV 1

/ 2 3 m m = 1 GeV 14 14 3/2 10 10 GeV GeV

15 15

(GeV) (GeV) / 2 3 = 1MeV 1 m ). PQ PQ = 10 = 10 =100 keV 13 13 V V I I

3/2 2.2

s s

10 10 m

/ 2 3 = 10keV 100 m + 12 12 CDM WDM (section 10 10 CDM WDM 2.1 = 600 GeV. The top (bottom) row is for 11 11 s 6 5 4 9 8 7 10 10 10 16 15 14 13 12 11 10 m

10 10 10 10 10 10

10 10 10 10 10 10 10 10

R R ) ( ) ( GeV T GeV T 16 16 – 30 – 10 m = 1 GeV 10

3/2 and there is a large set of parameters that lead to

= 10 MeV I = 1 TeV and m3/2 s 15 15

m3/2 = 10 MeV µ 10 10 m3/2 = 1 GeV = GeV discussed in section 11 from axino UV and gravitino UV production. We ﬁx 1 . M 10 14 m = 10 keV 14 Low Scale mediation: DFSZ 3/2 = 0 10 10 = 10 keV )10

3/2 2 2 = PQ PQ m . h / ( (GeV) (GeV) 2 =V =V I I s s PQ PQ & M 13 13 V V R 10 10 T 12 12 CDM WDM = 2, and 10 10 CDM WDM β . Contours of Ω enter the light gray excluded region. It is worth noting that for clarity we only 11 11 15 4 7 6 5 9 8 10 10 10 10 13 12 11 15 14 16

10 10 10 10 10 10

10 10 10 10 10 10 10 10 = 4, tan

R R ) ( ) ( GeV T GeV T Acknowledgments We acknowledge useful conversations withHarigaya. Marcin Badziak, This Kimberly Boddy, work and was Keisuke supported in part by the Director, Office of Science, Office of the cosmology with figure show the prediction for onecollider value signals of with viable cosmology. Figure 15 D JHEP03(2017)005 16 10 = 4, (A.1) D , which 15 i 10 +γ/Z/h +Z/h of the PQ (red) or to ˜ ˜ i = 2, q →a →G 14 1 1 µ h/Z q ˜ ˜ N N 10 + GeV a ), together with its 13 15 (GeV) 10 R 1.1 = 10 T I CDM WDM . We fix s ). 12 15 10 as small as possible. Within 2.1 | i = 1 MeV q = 100 keV | 11 3/2

3/2

m =

1GeV 1 m

/ 2 3 m (section 10 + Neutralino LOSP 3 2 6 5 4 1 2.2 PQ 10 -1 A 10 10 10 10 10 10 V

= 600 GeV. The left (right) panel is for the

) ( τ m c q s ˜ m . We normalize the PQ charges a 10 11 contours of ﬁgure . – 31 – exp 10 i as explicitly given in eq. ( v = 0 A 9 = 2 i 10 h +γ/Z/h +Z/h Φ ˜ ˜ = 1 TeV and →a →G 8 1 1 µ

= 1 GeV 10 ˜ ˜ N N and the axino ˜ GeV discussed in section = 3/2 s GeV

m 10 1 7 15 (GeV) M 10 )10 R = 10 T & I ( s 6 2 = . / 10 = 1 MeV 2 = 100 keV R Low Scale mediation: DFSZ 3/2 M T CDM WDM = 100 MeV 3/2 m ﬁlls the supermultiplet m 5 3/2 a 10 m . The lifetime of the neutralino LOSP, decaying dominantly to ˜ (pink), predicted from the Ω 4 = 2, and 3 2 5 4 6 1 10 -1 γ/Z At energies below the PQ breaking scale the theory is still approximately supersym- 10 10 10 10 10 10

1 + N

) ( τ m c ˜ ˜ The axion superpartners, the saxion breaking fields in such a way that they are all integers and the next appendix, we use these results to computemetric. the We decay assume widths the used PQwe in symmetry expand this to around work. be their broken vacuum by expectation a values set (vev) of chiral superfields Φ Physics, which is supported by National Science Foundation grant PHY-1066293. A Axion supermultiplet interactions In this appendix weaxion develop supermultiplet. a We general consider framework both self-interactions to and describe couplings effective to interactions MSSM fields. of the In AC02-05CH11231 and by the National SciencePHY-1316783. Foundation under R.C. grants was PHY-1002399 supported and Research in Fellowship part under by Grant the No.partment National of Science DGE Energy Foundation 1106400. grant Graduate numberin F.D. DE-SC0010107. part is at The supported the work by of Institute the L. for U.S. J. Theoretical De- Hall Studies was ETH performed Zurich and at the Aspen Center for G tan cosmology with High Energy and Nuclear Physics, of the US Department of Energy under Contract DE- Figure 16 JHEP03(2017)005 µ (A.8) (A.5) (A.6) (A.7) (A.2) (A.3) (A.4) . The α W . However, ..... A ), we identify )+ † A.4 G A .... + + A ( d , consistently with the † A H . A † u c term. . A is canonically normalized. + 2 µ i A v AH 3 A PQ + h 3 i V q α PQ µ . W i V α X µ µν q . ˜ 1 2 . G . + + 2 i µν PQ PQ d v θA W A G V 2 i 2 † H q d a u iαV a A f 2 i and the QCD field strength on the axion superfield is the following Z DW + 2 = X √ µH a 2 3 π N α A g – 32 – = PQ = 32 v DW d † = 2 2 → N PQ = π A H 2 3 a ). The axion superfield V u ˜ PQ g f G A 32 + V H A.3 − aG A and the supersymmetric QCD field strength ) to be canonically normalized, and the Kählerpotential L = A i PQ A q A.1 V µ q appearing in the above expression is the QCD anomaly coefficient AW W exp can appear in the superpotential of DFSZ theories, where the L 2 i DW v exp A N as the model-dependent PQ charge of the µ i X µ = q = i W Φ † i Φ i X Holomorphy and PQ invariance forbid superpotential self-interactions for = K term is PQ charged. We are allowed to write the PQ invariant operator Here, we denote This function only depends oninvariance the under PQ the invariant combination shift in eq. ( the axion superfield A.2 Supersymmetric interactions We assume the fieldsreads in eq. ( Upon expanding in component fields the supersymmetricthe expression effective in interaction eq. ( between the axion domain wall number of the PQ symmetry. Finally, we define the axion decay constant The PQ symmetry is broken by a color anomaly, which generates the low-energy interaction between the axion superfield The interactions ofinvariance. the axion supermultiplet are significantlyA.1 constrained by this Color shift anomaly The effect of a PQ rotation with an angle this convention, the effective PQ breaking scale is defined as follows JHEP03(2017)005 (B.1) (A.9) (A.12) (A.10) (A.11) . ..., + † ˜ a ˜ a † term necessary for a a a )˜ 2 / + ˜ 3 Bµ (e.g. sequestering). ˜ a , , m ˜ a d † H ˜ a AAX Pl u X c mess ). As discussed in the main text M H Pl + M s m M X F, 3 PQ PQ A 2 mess V 2 κ V √ AAX ( 2 c µ † M q 3 √ F A F/ ˜ 2 η √ ], and therefore we have the upper limit on + + √ = (2 a 17 exp is the associated goldstino. 2 1 2 A µ ). Once we expand in component fields, we find + / – 33 – 3 η − θ X a∂ m A.7 µ B = c AAX ˜ a = s ∂ − c ˜ a X = = . A consequence of this operator is a contribution to the PQ V mess κ F Pl 2 M M √ SUSY SUSY . − K W AAX = c mess − ˜ a ˜ a M = saa,s . L SUSY Bµ the mass scale of the particles coupling the two sectors. We expect this operator θK = 4 is the SUSY breaking scale and ˜ d F mess F Z B c M Furthermore, we also require the presence of an effective SUSY breaking is transmitted to the PQ sector through the higher dimensional oper- first two processes arethe mediated cubic by Kählerpotential the term in axion eq.the multiplet ( operators self-interactions, more specifically Finally, neutralino decays tocompute the the axino decay or widths gravitino for lead all to these displaced processes,B.1 collider using the events. interactions We Saxion derived above. decays The saxion can decay to three possible final states: axions, axinos and Higgs bosons. The B Decay widths Saxion decays are responsibleDecays and for inverse reheating decays of thedance. neutralinos universe The and and charginos axino generate producing decay an to dark axino the radiation. freeze-in gravitino abun- creates a warm DM population and vice versa. with where we have used theof known this relation work, it is notis natural some to good have an reason axino to much suppress lighter than the the size gravitino, ofsuccessful unless the electroweak there symmetry coefficient breaking to be generated bythe Planck mediation scale scale dynamics [ axino mass ator with Finally, we account forfield SUSY breaking. We find it convenient to employ the non-linear where A.3 SUSY breaking interactions JHEP03(2017)005 . ) ∗ i A H 2 (B.5) (B.6) (B.7) (B.8) (B.2) (B.3) (B.4) iσ = (for more ) c . β ( i ) . H c , and in such a . . Z ) providing elec- 0 m ! + h G d iA 2 − H ) and one CP-odd ( + is non-zero, axinos can √ A u and with H and H H κ d H . For models with only a m (

/v 2 2 PQ G u s = , , and , which takes the electroweak v √ /V ) as well as the SUSY breaking h 2 i . = SM ) v PQ c B 3 H ( i β Heavy Heavy A.8 2 Heavy V PQ . q , i V 4 µ s s q µ β H β H m 3 s P 2 2 µ 2 PQ PQ ˜ 2 a q + πm m = m 2 πV πV d 16 2 8 κ κ + sin + cos ]) 64 κ H D ) † d ). We consider the decoupling limit where the c 26 ( SM SM = = – 34 – H = is a free parameter. If ˜ a H + ˜ aa a κ β H β H ,H u → → s s visible H ! † u Γ Γ ). The resulting scalar potential interactions are 0 → = 8 for the full MSSM Higgs sector. contain three Goldstones ( H s 2 = sin = cos iG d Γ D √ + d u + s H h A.12 H G H lives within the multiplet 2 + PQ h µ v V and µ

q u 2 H = √ = 4, whereas SM = H D sHH V = 1. In more general cases κ counts the number of kinematically allowed final state particles. For a SM Higgs D More importantly for the reheating process, the saxion can decay to Higgs bosons and Here, sector we have where we introduce the ratioWe between report the here two the vevsgeneral tan decay expressions width see in appendix such A a of decoupling ref. [ limit and for large tan symmetry breaking vev. Thelimit decoupling the limit connection holds between gauge as eigenstates long and as mass eigenstates reads non-SM fields are heavy. In such a limit, we find it convenient to introduce the doublets The SM-like Higgs boson The Higgs doublets troweak gauge bosons longitudinalneutral modes, scalars two and CP-even one ( charged scalar ( longitudinal electroweak gauge bosons. Theseric decays interactions are arising mediated from from theinteractions the supersymmet- arising superpotential from in eq. eq. ( ( single PQ breaking field, orwe theories have with more thanbe one but copiously all produced with from thewidths same are saxion PQ decays. charge, Neglecting the final state masses, the decay Here, we define the dimensionless parameter JHEP03(2017)005 (B.9) ] (B.18) (B.19) (B.14) (B.15) (B.16) (B.17) (B.10) (B.11) (B.12) (B.13) 59 , , , , . , , , 2 | 2 2 i 2 | | ! 2 | i i i 4 . 2 V = 1) = 1) 4 = 1) − − d V R β † † † R a h e αβ c β β f W β | c c F c | | | †

RR VV + UU a + + 2 + λ | = = = = , 2 2 µ i 2 | | 2 | i i σ − i R 3 V 2 U U 3 e † † † ψ 0 u ., U αβ R ). Likewise, for charginos β R R c h e V U β σ . s β ( β ( ( | s s s 0 d | | | − h e ). B.10 + h † i i is achieved through the rotation 3 π φ e ˜ a π π µ e ˜ a C π N i ν ˜ 32 f G W m m m 32 16 f B.13 N m 16 – 35 – µ 2 2 χ ∂ e 2 2 B J µ σ Pl µ µ , ν µ µ i PQ PQ µ µ = PQ PQ µ µ σ M ) and ( q q V V q q ! , V V , 2 2 e χ + + − − √ + u e e ψ ψ h e B.12 f W = = † = = † = = e + +

V U N, ˜ µ H G φ i ˜ aH ˜ aφ e R L J = i N = = e C → → + = → i + − e e → ψ i e e ˜ N a e C C χ e ˜ C a Γ Γ Γ Γ defined in eqs. ( V is the rotation matrix defined in eq. ( and R U For neutralinos decays and inverse decays we have Gravitino decays are mediated by the Planck suppressed supergravity operators [ with B.3 Gravitino decays where and as expected wepositively find and that negatively the charged mass mass eigenstates eigenvalues fill for a the Dirac two fermion. arrays are the same. The containing positively and negatively chargedtwo states, chargino respectively. arrays We separately rotate the and the connection with mass eigenstates Likewise, we group the charginos into the two different two-dimensional arrays We start by definingcollect the the mixing neutralinos matrices into the for four-dimensional the array MSSM neutralinos and charginos. We B.2 Neutralino and chargino decays to axinos JHEP03(2017)005 ]. ), ), Pl 61 , M B.18 60 5 (B.22) (B.23) (B.24) (B.25) (B.20) (B.21) 3 √ ( ) as well as F/ = φ, χ 2 / 3 m , a αβ F † ) supermultiplets. We note that a , . . λ 2 µ 2 Pl 2 Pl V, λ Pl . σ 2 2 / M ] and the relation 2 / is the field strength of the vector field 3 3 ˜ 2 η / 3 3 αβ / ˜ 5 a π M 3 µ π M σ 2 3 m 63 m F ∂ , since all other R-odd particles produced , ., m m 2 32 1 c 1 384 . ) 2 3 e 62 √ N ) π m V φ ( c r ( c − 96 + h i N † N – 36 – µ φ = ) and vector ( ). Here, = J ν = → µ ˜ G a µ V λ φ, χ χ ∂ φχ V, λ 1) for the final state (gluino, wino, bino). ˜ η ∂ ˜ µ → → G , → σ ˜ 1 3 ˜ F G ˜ a G ν , Γ Γ Γ − σ = 3). Likewise, the decay width to gauge multiplet compo- = = ˜ η µ L J contains both chiral superfields with components ( (quarks) c µ N J The last cases we discuss are the ones relevant for displaced collider signatures. We The first case we discuss is the axino decay to axion and gravitino. The associated An important exception is when the main source for the soft masses is Anomaly Mediation [ 5 . The gravitino decay width to chiral multiplet components reads only consider decays of the lightest neutralino For the light gravitinos considered here, these corrections are very suppressed. axino decay to saxion andthe gravitino general is supercurrent assumed result, to and be accounting kinematically for forbidden. only Upon the using axion final state, we find Upon using the Goldstino equations ofwe motion recover [ the interactioncurrent. between the Goldstino and the flat-space global SUSY super- where we consider again boththe chiral above ( interactions canby also identifying be the derived longitudinal from component the of full the supergravity gravitino result in eq. ( approximate the process withequivalence theorem. decays The to process longitudinal is gravitinos, correctly described in by accordance the with effective Lagrangian the Here, the multiplicity factor is (8 B.4 Axino andIn neutralino decays the to last gravitinos partWe of assume this the appendix, gravitino to we be consider much decays lighter to than final the states decaying involving particle, gravitino. so that we can where we neglect thequark/squark pair final ( state massesnents results and in the color factor accounts for a decay to a gauge superfields with componentsV ( The supercurrent JHEP03(2017)005 ) ) . B.6 (C.3) (C.1) (C.2) NLSP (B.26) (B.27) (B.28) (B.29) τ ( a 2 LSP = m τ − a NLSP m 2 2 NLSP m , , , . 1 sec). . , is related to the gauge 1 2 2 Pl 2 Pl 2 & 1 2 Pl da . 2 LSP Pl − e M ) # N M M m 2 M 1 2 a 2 1 / 2 1 2 e ( 1 / 5 N e 2 3 / / e 5 N + 2 3 e v 5 N 2 3 5 N NLSP NLSP 2 3 τ m τ a m eq m a are the time and scale factor at m m a τ π m 2 π m a π m π m 2 NLSP eq Z 48 96 LSP a 48 τ LSP 96 m 2 2 p | eq 2 m | 2 2 eq | t | w β a β w is a (nearly) massless axion. We always c 2 c follow from four-momentum conservation in the decoupling limit as in eqs. ( and c s = 21 a 31 = 31 21 h } R eq 1 + R ) t R R ). For decays to photons we are only sensitive " dt − NLSP − + ) + τ – 37 – = t w β β ) ( NLSP w s s ) t B.10 s ) τ c ( , where a ( 11 41 a ( a 41 a ( 11 LSP R R v | | R R | LSP | LSP LSP = = , p = LSP p = E eq ) t L NLSP T τ ˜ → ˜ G h G γ Z ˜ ˜ ) = GZ GZ NLSP a → → = ( τ → → 1 1 ) after decays is just a consequence of free streaming. The initial ( e 1 1 e N a N FS e e ( N N Γ LSP Γ λ v LSP Γ Γ v E { = } is the NLSP lifetime, whereas τ LSP , p ) , justified if NLSP decays happen after BBN (i.e. NLSP 2 τ τ LSP / B.7 1 E The LSP velocity t { ∝ The LSP momentum red-shifts with the Hubble expansion a energy and momentum at the decay time Here, matter-radiation equality, respectively. Inand the we changed second integration variable equality, by using we the defined relation for a radiation dominated universe Our framework provides a warmand dark matter subsequent source decay through NLSP productionassume of that NLSP particles if theIn LSP this is appendix, the we gravitinocomponent provide the (axino), calculation then of the the NLSP free is streaming the length axino for (gravitino). such a warm where we identifyand the ( SM-like Higgs boson C Free streaming of warm DM component Finally, for decays to Higgs bosons we have For decays to Z bosons, we have both longitudinal and transverse final states eigenstates through the rotation into eq. Goldstino ( interactions with the neutral gauginos, and we find at collider will promptly decay to it. The mass eigenstate JHEP03(2017)005 × 2 . (C.8) (C.4) (C.5) (C.6) (C.7) 2 NLSP . ' m . The free eq !# t Phys. Rev. 2 , / LSP 1 Phys. Rev. Lett. , /m Sov. J. Nucl. ]. . In the , (2011) 043509 2 sec / τ LSP 4 1 93 Mpc, p NSLP ) τ τ . 10 . SPIRE eq by identifying the scale a ' D 84 IN   , 2 [ ' t/t FS eq ( λ )] LSP ) eq /a τ NR a a LSP . a τ LSP NLSP m a eq ( ( t p m LSP m 10 ) = ]. τ LSP ]. m NR NR Phys. Rev. eq t a p − F

(1978) 223 a ( a , ) a ≥ eq ]. 40 a a < a SPIRE SPIRE 1 log ( . 1 + ) can be approximated as follows IN IN ) results in F [ [ and we find s SPIRE C.3 1 + log ) C.1 + 1 + 0 IN /a a eq " ) [ LSP – 38 – 1 A simple solution to the strong CP problem with a a a 2 NR ( ( m invariance in the presence of instantons / LSP LSP ), approximately reads a 1 τ LSP LSP t p m ' m ( (1981) 199 LSP LSP m (1980) 497] [ C.1 ) p Phys. Rev. Lett. p τ eq ' sec and a , a   31 eq 4 eq NR ) p t NSLP a Constraints imposed by CP conservation in the presence of ), which permits any use, distribution and reproduction in CP conservation in the presence of instantons a a eq (1977) 1791 τ ]. ]. eq ( ( 10 t B 104 a v 2 = 2 LSP ]. ' p ) = log D 16 SPIRE SPIRE a FS Yad. Fiz. ( FS LSP IN IN λ λ [ F ][ NLSP m SPIRE CC-BY 4.0 On possible suppression of the axion hadron interactions IN m Phys. Lett. 10 [ , This article is distributed under the terms of the Creative Commons A new light boson? , correspondent to the time when the free streaming LSP enters the non- Problem of strong Phys. Rev. Dark matter in the Kim-Nilles mechanism , NR (1980) 260 [ by imposing (1978) 279 a 31 6 Mpc NR . 40 a 0 (1977) 1440 limit, the free streaming length reads ' arXiv:1104.2219 harmless axion Phys. [ instantons Lett. 38 sec, and the time dependence of the scale factor It is convenient to derive an approximate expression for A.R. Zhitnitsky, E.J. Chun, S. Weinberg, F. Wilczek, M. Dine, W. Fischler and M. Srednicki, R.D. Peccei and H.R. Quinn, R.D. Peccei and H.R. Quinn, FS 12 LSP λ [6] [7] [3] [4] [5] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. We evaluate this expression10 by using them known values We find streaming length, as defined in eq. ( factor value relativistic regime. The LSP velocity in eq. ( and the free streaming scale defined in eq. ( JHEP03(2017)005 ]. ] , Phys. B ]. , (1984) SPIRE (2012) (2001) ]. IN ]. ]. (1979) 103 ][ 05 SPIRE 43 D 85 IN B 138 SPIRE SPIRE SPIRE ][ IN Phys. Lett. IN ]. IN ]. ]. hep-ph/9601367 JHEP , ][ ][ [ ][ , ]. SPIRE ]. SPIRE SPIRE Phys. Rev. IN IN Phys. Lett. , IN (2010) 036 [ , hep-ph/0005116 ][ SPIRE ][ [ IN Phys. Rev. Lett. SPIRE 06 (1996) 3494 , Axion cosmology with its scalar ][ IN ]. hep-ph/0005123 ]. ][ [ 76 Freeze-in production of FIMP dark ]. hep-ph/9510461 arXiv:1106.2452 arXiv:1302.2143 JHEP (1980) 493 [ Freeze-in dark matter with displaced [ [ SPIRE , Experimental signatures of low-energy SPIRE IN Can confinement ensure natural CP Axinos as dark matter IN SPIRE ][ (2000) 095008 After primordial inflation ][ IN B 166 ][ arXiv:1506.07532 arXiv:1603.04439 [ (1996) 313 (2001) 023508 [ hep-ph/9803263 (2011) 065 D 62 (2013) 047 [ Can decaying particles raise the upper bound on Production of massive particles during reheating hep-ph/9407248 Largest temperature of the radiation era and its Phys. Rev. Lett. – 39 – 08 [ 12 , D 64 Supersymmetric axion grand unified theories and their B 383 ]. Nucl. Phys. The cosmological axino problem Is it easy to save the gravitino? (2015) 024 , JHEP hep-ph/9809453 (1998) 44 JCAP Effective interactions of axion supermultiplet and thermal Phys. Rev. , ]. [ , , arXiv:0911.1120 Solving the cosmological moduli problem with weak scale 12 SPIRE (2016) 075001 (1995) 229 hep-ph/0106249 [ ]. IN [ Phys. Rev. ][ Dark radiation and dark matter in supersymmetric axion models B 437 SPIRE Phys. Lett. , ]. IN , D 94 JCAP problem SPIRE B 449 , ][ IN µ ][ (2010) 080 (1999) 063504 SPIRE IN 03 Thermally generated gauge singlet scalars as selfinteracting dark matter [ (2002) 091304 Phys. Lett. ]. Collider signals from slow decays in supersymmetric models with an intermediate Thermal production of axino dark matter , Phys. Rev. D 60 , 88 Weak interaction singlet and strong CP invariance Nucl. Phys. ]. ]. , arXiv:1104.0692 JHEP [ SPIRE , IN hep-ph/0101009 (1983) 30 [ [ SPIRE SPIRE IN IN arXiv:1003.5847 gauge mediated supersymmetry breaking [ scale solution to the cosmological implications the Peccei-Quinn scale? predictions with high reheating temperature Phys. Rev. superpartner [ invariance of strong interactions? 015008 inflation 127 265 production of axino dark matter Rev. Lett. 033 [ signatures at colliders matter S. Dimopoulos, M. Dine, S. Raby and S.D. Thomas, S.P. Martin, M. Kawasaki, T. Moroi and T. Yanagida, R.T. Co, F. D’Eramo and L.J. Hall, P. Graf and F.D. Steffen, D.J.H. Chung, E.W. Kolb and A. Riotto, G.F. Giudice, E.W. Kolb and A. Riotto, J.E. Kim, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, C. Cheung, G. Elor and L.J. Hall, L. Randall and S.D. Thomas, M. Hashimoto, K.I. Izawa, M. Yamaguchi and T. Yanagida, M. Yu. Khlopov and A.D. Linde, K.J. Bae, K. Choi and S.H. Im, L.J. Hall, K. Jedamzik, J. March-Russell and S.M. West, R.T. Co, F. D’Eramo, L.J. Hall and D. Pappadopulo, D.V. Nanopoulos, K.A. Olive and M. Srednicki, L. Covi, H.-B. Kim, J.E. Kim and L. Roszkowski, A. Strumia, S. Weinberg, unpublished. J. McDonald, [9] [8] [27] [28] [25] [26] [22] [23] [24] [20] [21] [17] [18] [19] [15] [16] [14] [11] [12] [13] [10] JHEP03(2017)005 , ] ]. ]. , GUT M (2000) SPIRE SPIRE ] (2007) ]. Cold dark IN IN ][ (2016) 031 ][ B 494 Phys. Rev. , D 75 01 SPIRE IN ]. arXiv:1104.4807 ][ [ (2011) 115021 JCAP (2011) 039 SPIRE , constraints on warm and Phys. Lett. (2005) 30 (2014) 082 IN , α Phys. Rev. 04 arXiv:0802.2487 (2014) 12249 ][ D 84 , [ 10 arXiv:1104.1262 ]. arXiv:0812.0010 (2011) 027 [ ]. [ 112 B 619 JCAP Lyman- 06 , JCAP SPIRE , SPIRE IN arXiv:1604.01489 IN Phys. Rev. ][ [ , ][ JHEP (2009) 012 (2008) 123524 , ]. Phys. Lett. (2011) 123521 , 05 Cosmological aspects of inflation in a astro-ph/0402490 Cosmological constraints on the light stable Thermal leptogenesis and the gravitino problem He [ D 77 3 Constraints on mixed dark matter from anomalous SPIRE Coupled Boltzmann computation of mixed axion IN D 83 – 40 – [ Effects of unstable particles on light-element JCAP Hadronic decay of late-decaying particles and big-bang Proc. Nat. Acad. Sci. , (2016) 023522 , (2005) 7 ]. Solving cosmological problems of supersymmetric axion astro-ph/0402344 arXiv:1008.1740 Gravitino freeze-in Phys. Rev. D 94 ]. [ [ Thermal production of gravitinos , Leptogenesis, gravitino dark matter and entropy production ]. ]. Phys. Rev. (1993) 289 SPIRE ]. ]. ]. ]. ]. Structure formation in a mixed dark matter model with decaying , B 625 Solving the gravitino problem by axino IN SPIRE ][ keV X-ray line and the galactic substructure SPIRE IN SPIRE Some necessary conditions for allowing the PQ scale as high as 5 SPIRE SPIRE SPIRE SPIRE SPIRE . IN IN B 303 ][ 3 IN IN IN IN IN Phys. Rev. [ ][ , ][ ][ ][ ][ ][ (2004) 063524 (2010) 115029 Phys. Lett. , Did something decay, evaporate, or annihilate during big bang nucleosynthesis? Cosmological constraints on the scale of supersymmetry breaking D 70 D 82 Phys. Lett. ]. ]. (1982) 1303 , hep-ph/0701104 [ 48 hep-ph/0006211 [ SPIRE SPIRE arXiv:1306.0913 arXiv:1412.1592 astro-ph/0503023 arXiv:1103.4394 arXiv:1406.4138 IN arXiv:1012.3769 IN strong lens systems matter: controversies on small[ scales on warm-plus-cold dark matter models sterile neutrino: the [ nucleosynthesis abundances: lithium versus deuterium[ and 075011 [ Phys. Rev. [ Phys. Rev. gravitino in SUSY models with[ an axino or neutralino LSP neutralino dark matter in the SUSY DFSZ axion model 297 in the Asaka-Yanagida axion/axino[ dark matter scenario Lett. models in inflationary universe [ supersymmetric axion model A. Kamada, K.T. Inoue and T. Takahashi, D.H. Weinberg, J.S. Bullock, F. Governato, R. Kuzio de Naray and A.H.G. Peter, A. Boyarsky, J. Lesgourgues, O. Ruchayskiy and M. Viel, A. Harada and A. Kamada, M. Kawasaki, K. Kohri and T. Moroi, J.R. Ellis, K.A. Olive and E. Vangioni, V.S. Rychkov and A. Strumia, C. Cheung, G. Elor and L. Hall, K. Jedamzik, J. Hasenkamp and J. Kersten, T. Moroi, H. Murayama and M. Yamaguchi, H. Baer and A. Lessa, K.J. Bae, H. Baer, A. Lessa and H. Serce, T. Asaka and T. Yanagida, H. Baer, S. Kraml, A. Lessa and S. Sekmen, M. Kawasaki and K. Nakayama, M. Kawasaki, N. Kitajima and K. Nakayama, S. Weinberg, [45] [46] [43] [44] [41] [42] [38] [39] [40] [36] [37] [34] [35] [32] [33] [30] [31] [29] JHEP03(2017)005 ]. ] 08 , Phys. SPIRE B 212 , 06 IN ] , ][ JHEP , (2016) 131 ]. JHEP Mon. Not. Roy. , , 820 , Tech. Rep. Nucl. Phys. collisions at SPIRE , arXiv:1002.1967 [ IN pp (2016) A58 ][ hep-ph/0507150 (2005) 063510 [ arXiv:1507.03224 SuperWIMP solutions to 591 arXiv:1407.5066 , [ (2016) L23 Yang-Mills theories with local ]. Astrophys. J. (2010) 073 D 72 , 827 science book, first edition 03 results. XIII. Cosmological 4 SPIRE S ]. IN (2005) 181301 (2014) 3174 ]. ][ 2015 JHEP arXiv:1502.01589 95 , Phys. Rev. [ CMB- SPIRE , IN Astron. Astrophys. C 74 SPIRE , Astrophys. J. ][ IN Planck , – 41 – ][ ]. ]. New detectors to explore the lifetime frontier The two faces of anomaly mediation Anomaly mediation from unbroken supergravity The spectrum of goldstini and modulini Goldstini (2016) A13 SPIRE SPIRE CDM cosmology: simulating a realistic population of IN IN An assessment of the “too big to fail” problem for field dwarf Λ 594 Phys. Rev. Lett. arXiv:1504.01437 Eur. Phys. J. ]. ]. ][ ][ , [ , ]. TeV SPIRE SPIRE arXiv:1606.06298 Squark and gluino production cross sections in SPIRE arXiv:1307.3251 [ IN IN 100 [ IN ]. [ ][ ]. ][ ]. The unexpected diversity of dwarf galaxy rotation curves (2015) 3650 and Dark matter from early decays Concept for a future super proton-proton collider Future circular collider study — hadron collider parameters SPIRE 33 Breathing FIRE: how stellar feedback drives radial migration, rapid size 452 (2017) 29 Astron. Astrophys. , arXiv:1104.2600 IN arXiv:1202.1280 , [ [ [ (2013) 125 collaboration, K.N. Abazajian et al., 14 collaboration, P.A.R. Ade et al., ]. ]. ]. , 09 B 767 = 13 SPIRE SPIRE SPIRE s IN IN astro-ph/0507300 IN arXiv:1511.08741 arXiv:1512.01235 arXiv:1602.05957 (2011) 115 (2012) 151 JHEP [ arXiv:1610.02743 supersymmetry: Lagrangian, transformation laws(1983) and 413 super-Higgs effect FCC-ACC-SPC-0001, CERN, Geneva Switzerland, (2014). Planck parameters CMB-S4 √ Lett. [ [ small scale structure problems [ Astron. Soc. galaxies in view of[ baryonic feedback effects fluctuations, and population gradients[ in low-mass galaxies Reconciling dwarf galaxies with satellites around a milky[ way-mass galaxy C. Cheung, F. D’Eramo and J. Thaler, F. D’Eramo, J. Thaler and Z. Thomas, C. Cheung, Y. Nomura and J. Thaler, E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, F. D’Eramo, J. Thaler and Z. Thomas, A. Ball et al., C. Borschensky et al., J.P. Chou, D. Curtin and H.J. Lubatti, J. Tang et al., M. Kaplinghat, J.A.R. Cembranos, J.L. Feng, A. Rajaraman and F. Takayama, K.A. Oman et al., E. Papastergis and F. Shankar, A.R. Wetzel, P.F. Hopkins, J.-H. Kim, C.-A. Faucher-Giguere, D. Keres and E. Quataert, K. El-Badry, [63] [61] [62] [59] [60] [56] [57] [58] [53] [54] [55] [51] [52] [49] [50] [48] [47]