CORE JHEP04(2012)106 brought to you by by you to brought Springer - Publisher Connector - Publisher Springer by provided Springer is a natural Phenomeno- , March 7, 2012 April 20, 2012 March 29, 2012 : : August 31, 2011 : : g,h roduction in order to Revised dark matter produced Published Received Accepted 10.1007/JHEP04(2012)106 DFSZ models of the axion. doi: l, to include production via cal Physics, , s. First we discuss the definition [email protected] , Published for SISSA by , and Leszek Roszkowski e,f , Jihn E. Kim d Laura Covi, 1108.2282 a,b,c Beyond Standard Model, Cosmology of Theories beyond the SM, Axino arises in supersymmetric versions of axion models and

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E-mail: Gwangju 500-712, Korea National Centre for Nuclear Research, Ho˙za 69, 00-681 Warsaw, Poland Department of Physics andSheffield, Astronomy, S3 University 7RH, of U.K. Sheffield Busan 609-735, Korea Institute for Theoretical Physics, University G¨ottingen 37077 Germany G¨ottingen, Department of Physics andSeoul Astronomy National and University, Center Seoul for 151-747, Theoreti GIST Korea College, Gwangju Institute of Science and Technology, Asia Pacific Center forGyeongbuk Theoretical 790-784, Physics, Republic Pohang, of Korea Department of Physics, POSTECH,Gyeongbuk Pohang, 790-784, Republic of Korea Department of Physics, Pusan National University, [email protected] [email protected] b c e g d a h f Open Access logical Models, Supersymmetric Effective Theories ArXiv ePrint: Abstract: Ki-Young Choi, Keywords: thermally and non-thermally in light ofof recent axino development relative to lowThen energy we axion review one and for several refineavoid KSVZ the unphysical and computation cross-sections of and,SU(2) the depending dominant and QCD U(1) on p interactions the and mode Yukawa couplings. candidate for cold or warm dark matter. Here we revisit axino Axino cold dark matter revisited View metadata, citation and similar papers at core.ac.uk at papers similar and citation metadata, View JHEP04(2012)106 . . 4 6 7 1 4 a a 11 20 23 23 23 28 10 23 24 f ], an (1.1) g the 22 , 21 ]. supersymmetric 20 ry light axion field ken at some scale hermally, through out- e LHC were explored in ducing a light pseudoscalar, ing PQ charges [ a dominant component of cold onservation is absolutely stable and decays in the hot plasma, as , µν e G µν , can be a natural candidate for CDM if a G a s axino – 1 – πf α 8 = eff a L ]. The scenario was subsequently extensively studied durin 2 , 1 ] and implications for Affleck-Dine Baryogenesis in ref. [ ], with either a neutralino or a stau as the next-to-lightest 19 16 – – 3 17 , 10 – 8 The strong CP problem is naturally solved by introducing a ve 4.3 Structure formation 3.2 Non-thermal production 4.1 The relic4.2 density of dark Nucleosynthesis matter 2.1 The framework 2.2 The KSVZ2.3 model The DFSZ model 3.1 Thermal production Below the PQ scale, after integrating out heavy quarks carry The axion appears when the Peccei-Quinn (PQ) symmetry is bro of-equilibrium decays, or thermally, throughoriginally shown scatterings in refs. [ it is the LSP. The relic axino CDM can be produced either non-t The lightest supersymmetric particle (LSP)and with can R-parity contribute c to the presentdark Universe matter energy (CDM). density If as the strongan CP problem axion, is its solved by fermionic intro SUSY partner, an 1 Introduction 5 Conclusion A Details of axino relic abundance computation 4 Cosmological constraints 3 The production of axinos Contents 1 Introduction 2 Axinos effective axion-gluon interaction is given by last decade [ particle (NLSP). Ways ofrefs. testing [ the axino CDM scenario at th JHEP04(2012)106 a f red (1.3) (1.2) ) has 1.3 is the origin ] the PQ charges , its scalar partner 22 a , y for the SUSY version teraction eq. ( 21 esponding to the inter- will be discussed in the as a non-renormalizable have been obtained from eld carrying a PQ charge. gluon anomaly and this is a as obtained, without much f els [ e DFSZ models. For solving . ass was introduced by hand. bosons can be obtained in a le (after integrating out heavy ] -Srednicki-Zhitnitskii (DFSZ) ns in the KSVZ models while d source of uncertainty is the . In the DFSZ models instead α he exchange of massless gluons. le is limited to a rather narrow ϑϑ, ns, which have led to somewhat rd Model (SM) fields, depending containing the gluon field A W α F α W + W aϑ Tr [ 2˜ A √ is the field strength of the gluon field and ) can easily be supersymmetrized using the Z , for a Grassmann coordinate. 1. Different types of axion models have been a G a ) + 1.1 ϑ ] of the KSVZ and DFSZ classes of models since − – 2 – ia πf s 25 = 2 α + ], while for the DFSZ case precise constraints on √ s 2 ( 25 0123 2 − ǫ 1 √ ]. The supersymmetrization of axion models introduces = 2 = which contains the pseudoscalar axion GeV [ eff )[ A L 12 A 1.1 10 ] a simple insertion of a thermal gluon mass to regulate infra ∼ < 2 a f is its dual with ∼ < ] the PQ charges are assigned to the SM quarks. This difference µν 24 G GeV , 10 stands for an auxiliary field and 23 is the strong coupling constant, µνρσ ǫ , and their fermionic partner axino ˜ s A 2 1 s α F ≡ The effective axion interaction of eq. ( The couplings arising in these two popular classes of models Very light axion models contain a complex SM singlet scalar fi The axino as a candidate for CDM was originally studied mostl µν e astrophysical and cosmological considerationswindow and 10 the sca Effective axion multiplet interactionssimilar with way. the other Axino gauge production from QCD scatterings due to in where where different numerical results. Thequestion main of how technical to difficulty regulate an theIn infrared the divergences due original to t study(IR) [ divergences was usedcontrol and over the the leading subleading logarithmic finite term piece, w since the thermal m been considered in the literature in different approximatio superpotential of the axion and the vector multiplet a full axion supermultiplet next section. For the KSVZ axion, constraints on the PQ scale the strong CP problem, onegenerated needs by a a coupling heavy of quark theeffective axion loop coupling to when in the those the heavy KVSZthe fields models, coupling are appearing is integrated generated out by SM quark loops. are assigned to new heavyapproach quarks, [ while in theof Dine-Fischler different phenomenological properties [ in the low energy effective theoryfields) at the the electroweak gluon (EW) sca anomalythe term is Yukawa couplings the are source the of source all of interactio all interactions in th saxion G proposed, with distinctively different couplingson to their Standa PQ charge assignment. In the Kim-Shifman-Vainstein-Zakharov (KSVZ) class of mod of the KSVZ axionaction model, term in given an by important eq. production ( mode corr have not been derived yet. JHEP04(2012)106 ] ], 6 5 compared ˜ 2 τ ] gave a simple /m 2 τ 43 m ribution, in this paper ction. However in the th an emphasis on an was applied in ref. [ ing and the dependence ] a chiral transformation terms may have and in wo latter methods have teractions with unknown hand, are suppressed by 2 he axion SU(2) anomaly axionic models, the chiral mation but we improve it of order f Yukawa-type interactions no to the gauge multiplets of the DFSZ axino as dark not gauge invariant either, uction. Recently, in ref. [ n rate by less than 1%. In ndance are. In the previous variant, includes more terms . We shall come back to this le, and examine its effect on way when supersymmetry is g temperatures, which is the hich are independent of the ref. [ hen the axion SU(2) anomaly uon whose thermal mass can e general axion SU(2) anomaly terparts, the sleptons and the dark matter. In our paper, we her at one loop or at tree level, he KSVZ model. Only recently ds. It is also not clear if it is hermal mass and more control he high energy region where the tween supersymmetric particles ] a previously neglected dimension- 6 – 3 – is the largest mass in the loop. Therefore the error in M where 2 /M ] considered axinos in the DFSZ model. In particular, ref. [ 58 2 lepton m ] and [ 43 In view of the recent developments in estimating the QCD cont In the calculations published so far the couplings of the axi Moreover, in the present work we will also consider the role o Subsequently, a hard thermal loop (HTL) resummation method we also update and re-examine relic CDM axino production, wi approximate formulae for the relicstudy density axino of production light in axino thematter as DFSZ in model a and more the complete suitability way. of the left-handedinteraction lepton and doublets to leave was onlycoupling performed the re-appears axion to U(1) in remove contribution.fermion principle t T masses. from the Thethe leptonic corresponding ratio loops, axino w loops, on the other it is the onlyregime known important viable for method the at axinodiscussion relatively as in low cold more reheatin dark detail matter below. candidate other than the one of the gluon was also neglected. In fact, in their own advantages and limitations, andconvergence they of coincide the in perturbation t series is5 stable. term In in ref. the [ Lagrangian,was containing purely also interactions included. be our This paper, however changes we the adoptto axino the make productio original the way result of positive effective definite. mass Although approxi this method is allowing for a self-consistentover determination the of constant term the ina gluon the new t high calculation energy was presented regionof which, of the although axino not perturbative prod gauge series,be in in larger particular than the the decay gluino of and the axino gl mass taken together. The t neglecting the axion SU(2) anomaly is estimated to be at most refs. [ between the axino and the matterdepending multiplets, on which arise the eit model.particular how model We dependent willstudy, our investigate results only what for the effect the KSVZDFSZ axino these model model abu axino was production consideredon is the for dominated reheating the temperature by is the axino quite Yukawa different produ coupl from that in t in any case broken.slepton Rather and than saxion going masses as into parameters, theinteraction we will swamp and keep of here its th such SUSY in the counterpart, axino which abundance. is more tractab to the SU(3) term.rotation on However, in the supersymmetric leptonsaxion, extensions fields which of involves generates also additional theirjustified scalar couplings to coun perform for a those redefinition fiel in a fully supersymmetric JHEP04(2012)106 (2.1) ctive e KSVZ of the PQ denotes the V a f e, term is the PQ esult on the DM for the both the 1 c 1 , R T and constant pieces — The  2 re . i tive even for the invariant a d ino production and of sub- ntations of the KSVZ and f 5 orks in the following aspects: . In particular, our updated iγ nd U(1) interactions; i ld that can be reabsorbed into ¯ d  i i d reheating temperature as the maximum d m 5 d 2 γ c µ malised bath of SM particles starts. We can γ + i by a factor of 0.745. In comparing our results ¯ o the vacuum expectation value i d tional definition of the reheating temperature to d 1 u c 5 is the domain wall number. ˜ iγ + G, i i ¯ N u u i u – 4 – , which arise after integrating out all heavy PQ aG 5 3 a γ m c f µ u 2 2 3 c γ ] considers a reheating process from the decay of the inflaton where i c π 6  ¯ u and u 1 32 i , c V/N 2 X  + c we present our conclusions. a = we discuss several constraints on the scenario arising from a , we calculate axino production rate for the KSVZ and DFSZ f i 5 a 1 4 X f c 3 + ) a a µ f ∂ is usually omitted by absorbing it to the field strengths. Her ( ] low energy axion interactions were given in terms of the effe 3 g 25 , we define the axino by its relation to the shift symmetry of th 2 smaller than the gluon thermal mass; s can be defined to be 1 (if it is non-zero) by rescaling 3 c DFSZ models. leading terms in the massthese of the last gluon parts beyond ensure themass that logarithmic the cross-section remains posi neutralino and the staurelic as density the and NLSP relevant using structure the formation current data. WMAP-7 r In section We assume the instant reheating approximation and define the The strong coupling 1 2 (i) an inclusion of some previously neglected terms in the ax (ii) an explicit calculation of axino production via SU(2) a (iv) an update on the constraints on the reheating temperatu (iii) a derivation of the axino abundance in specific impleme easily translate the axino abundancemore given specific with this reheating conven scenarios. E.g., ref. [ temperature at which standard Big Bang expansion with a ther analysis of the axino CDM scenario improves on the previous w estimate of the uncertainties as well as on model dependence charged fields. The effective axion Lagrangian terms are and obtains a slightlywe smaller account abundance for of this axinos, difference. reduced where 2.1 The framework In a recent reviewcouplings [ with the SM fields axion models, and incosmology. section Finally, in section 2 Axinos and DFSZ axions. In section symmetry preserving derivative interaction of the axion fie so-called axion decay constantsymmetry breaking which scalar is field typically by related t JHEP04(2012)106 is a V (2.4) (2.5) (2.2) (2.3) in the es. They a V 6= 0, after = 2 i c 2 6= 0 arises from S h constant. This is 3 the axion. In both c 3 = c 4 terms, we have only ]. i odels corresponds to ]. An explicit model was 1 + 2 symmetry as 30 he axion emerges from c S 2 = 0 and 28 ion and is proportional h c . – , the SM fields also carry 2 3 PQ s f the axion multiplet, and ] 26 c S entioned earlier: the KSVZ u,d ctions of the saxion and the he massless field. σ t us consider the PQ sector ses is zero then the anomaly W scale. e axion. Hence the definition 29 (1) Q metry breaking. In the latter = axion-dependent PQ rotation U is a Yukawa coupling and iαQ 1 − c , Z , e in refs. [ a f ) 2 D a → a V φ, 2 D a − , while keeping 2 1 ), while in principle an infinite series of T 2 3 a ∗ c S + ,S φ 1 /f 1 2 S

S ( φ a and σ X Z – 5 – 2 g Z ∂φ ∂W c iαQ f

e = = a a → X , if it is non-zero, defines the QCD axion. Only this D 3 1 PQ c = carry respective PQ charges W + V 2 u,d c may have family dependence but here we suppress family indic Z,S = 1, and the DFSZ one to H 3 u,d → Q , and the anomaly interaction proportional to , c 1 Z are gauge singlet chiral superfields, = 0 ], and the first cosmological study was performed in ref. [ 2 2 S c 29 = arises. see table a 1 3 , and c . Hence the sum 1 | 3 a/f c term by a partial integration over on-shell quarks. For the Z,S + 2 2 In the following we will consider the two popular scenarios m Below we will examine more closely the KSVZ and DFSZ models of In the supersymmetric version of axionic models, the intera c c The supersymmetrization of axion models was first discussed In variant axion models, | 4 3 The potential in the global supersymmetric limit is integrating out the heavy fieldmodel, sector if responsible the Higgs for doublets PQ sym PQ charges, can be inserted if needed. the and the DFSZ classesthe of choice axion models. The KSVZ class of axion m terms in first constructed in [ with at high energy with the PQ breaking implemented as in ref. [ models one imposes theits PQ spontaneous symmetry symmetry at a breaking. high As energy a scale specific and example, t le of the axion at lowtherefore energy of must the be saxion connected and to the the axino, definition at o energies above the E connected to the well-known factbecomes that, unphysical if and one can of the be quark reabsorbed mas in the rotationaxino with of matter t are related by supersymmetry to those of th kept the lowest order terms (proportional to 1 SM quark loops.to The axion mass is given by the strong interact of the quark fields can shift the values of combination of the two couplings is physical, since a chiral where a parameter in the Lagrangian which determines non-zero VEV minimization process. The superfields transform under the JHEP04(2012)106 , 2 / 3 (2.7) (2.6) m vanishes. combine d SUSY ≫ u L c R R 0 Q q a U Q V + d and c R Q 0 L D + Q in the superpotential tion of the axion and ] for the superpotential L R 0 0 e supermultiplet can be q 29 Q dependently of the axion. on. f. [ a very small axino mass was u angian can then be obtained u can develop a non-zero value, and 0 Q sm is needed in order to deter- , Assuming that . H i a 1 − L V denotes collectively all the scalar Z S = 0 and SUSY is not broken then h Q Q , R φ ) f i d 2 d Q a Z Q 0 L h V = H − Q denote new heavy quark multiplets. In the − Q Q ) one has 2 R f , . σ m S 2 1 Q 1 R + Q 2.2 0 S 2 1 Q ZS ZS ( – 6 – PQ and − Z Z Z f f f L W ]. 2 combine to become a Dirac fermion with mass Q σ = = = 2 contains the axion field and can be identified with = 32 √ L . Q 1 2 √ 0 / 2 1 / Q Q ) ) ) we note that, if ∂Z 2 ∂S ∂S ∂W ∂W ∂W − 2 S KSVZ S develop VEVs and break the PQ symmetry. The fermionic 2.6 2 + W σ σ − S 2 1 1 Q Q S S S − − , and the PQ charge of the left-handed SM quark doublets d and = ( Q 1 = ( σ σ 1 S S + ′ S Q Q u S Q symmetry breaking takes place, as discussed above, 0 0 Z = ]. Recently the case of a direct coupling between the axion an and σ , while PQ 31 a ] Q Z V (1) 22 Z U , f is the gauge coupling of the gauge groups and 2 DFSZ KSVZ Model . The PQ charge assignment 21 g √ However, with soft SUSY breaking terms included, = Z Table 1 partners of the scalar partnerintegrated has out practically in a the supersymmetric same way. The mass low-energy and Lagr the whol After the to become a heavy Dirac fermion with mass In the KSVZthe approach gluon in fields, order one to introduces obtain the heavy the quark anomalous fields interac above, while another superpotential andgiven SUSY in breaking with ref. [ breaking sector was discussed in ref. [ 2.2 The KSVZ model both the axino and the saxion are mass degeneratein with which the case axi both theTherefore, a saxion full and specification the of axinomine the the become SUSY mass breaking massive, of mechani in the axino exactly. This was first studied in re m the axion multiplet. From eq. ( At the tree-level both DFSZ model 2 See text for more details. where as ref. [ fields. With the superpotential given by eq. ( JHEP04(2012)106 . 1 er ] 34 (2.9) (2.8) (2.12) (2.11) (2.10) (2.13) 2, and 5 √ / ) ) 2 ˜ g S µ − D µ 1 . S ˙ α ˜ ( gγ ]. i ¯ λ ]. ., 6 ≡ µ 25 SUSY invariant form, + 2 D S ˙ α a s can be obtained in the king superpotential [ µ α D r, in the limit of unbroken couple to the light quarks. matrices then reads. = 1 [ aly matching condition. In a + h.c.) 3 c ϑϑσ D . , 2 + h.c. λλ u + A a ! α − β H F a a , d ϑ W ], eq. (6.11)) ψ ψ a µν  2 ( H µν aα 2 G 33 1 2 from that in ref. [ e

√ = 0 and a µν G S 2 G = µν + P aµν c s β α f ˜ a )  G M , ϑA W Higgs doublets carry PQ charges and thus a = ν ( a G . The charge assignment is shown in table A s 2 is the number of the heavy quarks and we ¯ σ 1 + Q Y d a F µ c ˜ gD PQ 2 Q σ N ˜ a – 7 – , ϑ ( πf Z s PQ 2 N )) + 8 i (1) 2 α ! a ˜ g U(1) + W − 5 . U a are given by ([ − a γ iλ ¯ = πf s λ × ˜ g = i a µ 2 V 5 α − L ϑψ γ eff a α √ D ] 2 ˜ gγ

2 L β α ν ( = 2 W √ δ µ DFSZ − =  , γ ), including the axino interaction with the D-term has re- 2 a D + µ W ˜ g f = + γ and can be obtained after electroweak symmetry breaking (EWSB) + [ ia a α 2.11 2 eff e ] to calculate the thermal production rate of axino dark matt G A a µν a µν + iλ 6 √ L e s G − aG ˜ aG i = = aµν + a α is the pseudoscalar component of the superfield G A ( ), as a W a  2.8 a = 1 in our case. Then we have s α πf Q 8 N = eff The axion superfield The low-energy interactions of the saxion and the axino field L Our saxion-gluino-gluino interaction differs by a factor of 5 consider The effective Lagrangian ( where the gluino and axino 4-spinors are given by this way one finds that the axion anomaly term becomes where the axion by integrating out the heavy quark multiplet or by using anom The effective Lagrangian in terms of the Bjorken-Drell gamma through the coupling ofTo the this axion end, to one the adds Higgs a doublets non-renormalizable which term to the PQ brea 2.3 The DFSZIn model the DFSZ framework, the SU(2) The anomaly coupling cently been used inand ref. will [ be the basis also for the present analysis. same way by integrating outSUSY the the heavy (s)quark low-energy fields.including effective eq. Howeve Lagrangian ( should be given in a the axion decay constant is the light quarks are also charged under JHEP04(2012)106 1 ∼ 2 a (2.20) (2.18) (2.15) (2.16) (2.17) (2.19) (2.14) /V P ). They are µM ∼ 2.13 s the quark triangle f ), . . c µν 2.10 symmetry is broken, the e nd such uncertainties will E B ore the axion cannot have . d from eq. (    µν , y obtain corrections of order -term. PQ a a u 0 , µ LH V V 1 aB P τ c 2 u (1) u a a F 0 y v 0 v 2 1 z V U u 2 u ϑ + α πV . Note that here a . The lightest state, which we will gh the Higgs Yukawa couplings, which are 0 Q 8 c 0 Q β i u z V + ongly suppressed and we will neglect it here. D + Z    H 1 − − h d . , 2 α ˜ a d d d Z 2 v 2 ϑ f . P = H a d 2 ˜ a 2 aµν d 2 QH 2 0 Q v b ≃ − αβ √ − f z d √ W y ǫ √ aY Y 1    1 + symmetry, given by + Q a i i a µν ˜ + a anomalous interaction can be generated via a ≡ 2 – 8 – 1 2 ,C a 1 c 0 − = S S u V = Y PQ h h ia s aW U σ 2 and is given by 2 ˜ a H a u a a i d Q 0 + a √ 2 1 (1) aW W 0 4 z 0 V 1 z H S α πV C U Z σ s QH symmetries are both broken, the axion is identified with 8 q t f i h Q y − = 2 2 0 0 PQ z and U(1) S 1 = ). Since EW symmetry is not broken, the axion (and thus = ) and h = S L (1)    a 2.2 U Z 2.15 f MSSM ) and we have used ˜ Z anomaly W , . Instead, the axino mass eigenstate can be obtained from the L 2 2 ˜ a S , 1 a However, SU(2) is given in eq. ( anomalous interaction with gluons generated at one loop via c 6 PQ W Before the EW symmetry is broken but after the The superpotential including light quarks is given by After the EW and the An axino-gluino-gluon coupling can arise at two loops throu 6 non-vanishing even above EW symmetry breaking, but it is str which coincides with eq. ( Here we defined the component fields in the same way as in eq. ( higgsino triangle loop through the Yukawa coupling derived where and similarly for generates a phenomenologically acceptable supersymmetri given by the axino) do notthe mix SU(3) with the Higgs (and higgsino) and theref in the basis of (˜ diagrams. These anomaly interactions appear alsoof for one the from axino, SUSY butbe breaking, ma compared later to encoded the in axion the couplings, coefficients a mass matrix massless axion Goldstone boson is identified as identify with the axino, has mass the Goldstone boson of the broken JHEP04(2012)106 (2.21) (2.26) (2.24) (2.22) (2.25) (2.23) ), , ned through the axion u 2.14 . , /v β, u , , are the pseudoscalar fields iP ··· ··· re given by e u ··· tan . ··· u + P + l, eq. ( / d,u d h u u + , + v 2 P -boson and the orthogonal pseu- P P βa  βa β + ∗ 2 d d √ − = 1 Z 2 a . 0 v v u u a  + 1 V d a u v d 2 u u a cos cos , are given by 2 H 2 f u v d,u 2 2 a d a P v cos v v 2 V 1 v R β β v v u 2 d V = S H 2 d β + 2 2 − d i ) v u 2 v = v , while 0 v v 2 u u d L d − − i i v − u , + u P P sin  0 sin +  2 1 d 2 sin | ), we obtain ∗ t s u 2 u u 2 v a a 2 a S u y a v v P v ( v v V V V a a d = u / H 2 exp 2 exp − v | a V V 2 v 2.22 – 9 – v ) 0  = R V ,H v 0 = L u and h d − u d + + d u u u v q v v H s q q v v u s β, Q L d /v v R ··· ··· L a v d a u P u + . a u u + d ,H 2 iP + + 2 u 2 u V v u 2 = d u R v d e d v u v 2 v v d v √ a a u √ d v v √ ( v v − H t ∗ t a t = tan a u 2 u / h + y y y v  V v 2 V u d u = = = = 2 2 d d 2 2 d − v + 2 1 v v √ − v − v v v v a L d A = v = q = Z = ≃ ≃ L and d σ L u d = = 2, and we expanded the Higgs field as Q Q P P i 0 d v 0 √ | H / d ) 2 H | a 0 h − ) with the last term of eq. ( 1 a 2.20 2 = = ( √ s / d a v The neutral Higgs boson component eaten by the The interactions of the axion with the matter fields are obtai with contained in the electrically neutral component of doscalar Higgs, i.e. the phases of where Considering the Yukawa interaction from the superpotentia Equating ( the Lagrangian terms for the up-type quark axion couplings a part of the phase of the Higgs fields, up to a common normalization constant. JHEP04(2012)106 L ), ) at 2.1 (2.29) (2.27) (2.28) 2.27 , involves ng matrix . However R T d,u H ) and eq. ( ; compare eq. ( . At high reheating ions one obtains the u,d 2 R Universe with axinos. teraction terms to the c 2.26 e axino mass eigenstate T l parts of the axino mixes with the e above to the coefficients ss eigenstate is not exactly no production through the β. ndependent of symmetric Standard Model permultiplet, which remains urther, but point instead to . factor, as the QCD anomaly ture 2 s of particles in thermal equi- below the EWSB scale, with u (TP) depends on the reheating , ˜ h 2 u a u ) and SUSY breaking, which is of v µν v = sin 2 u /f u e G q 2 2 u Q Q 2.13 v v m µν + − = 2 d d aG v d 2 ) ˜ h 2 d a d terms given in eq. ( anomaly coupling is absent and the SU(2) v V Q N 2 c d s c (2 even above the EWSB scale. These tree-level + Q α π 2 a – 10 – a 8 − β, c V thermal production s 2 σ 2 = ˜ a µ/f Q a 4 V . σ q = cos Q /N 2 a 2 2 d ] and obtain for following values for anomaly v v V L ) and is given by 25 = = ˜ a ). Such a field is a good approximation to the physical axino ]. = 2 u 2 2.20 a c 36 f , 2.29 , is negligible. Otherwise the whole axino-neutralino mixi a 35 /f 0 z and a denote the fermionic components of the electrically neutra /f = 6 and again u,d N d,u v ˜ h After integrating out all the Higgs fields except the axion su In the DFSZ models, there are also axino tree-level Yukawa in In the supersymmetric limit, below EW symmetry breaking, th the state given in eq. ( since supersymmetry is broken,higgsinos in differently general than in the the axion DFSZ with models the Higgs and the ma Here in the definition of the axion [ and similarly for the down-type quarks. We can now compare th with a singlino LSP [ has to be consideredthe in related detail. studies We in will not the discuss case this of case the f Next-to-Minimal Super light, all the axion couplings arise from the quark and squark with a coupling of the order of only if the mixing generated fromorder the superpotential ( low energies above theaxion-anomaly quark interaction term. masses. It is At then one given loop by in the SM ferm can be read off from eq. ( where term has, and thusdecay they give and/or the scattering dominant processes contribution to at axi low reheating tempera the Higgs and higgsinointeractions with are coupling not suppressed by a gauge coupling or a loop 3 The production ofThere axinos are twoFirstly, efficient they and can be robust producedlibrium. ways through This of scatterings mechanism, and which populating we decay temperature call the after early inflation. The other mechanism, which is i anomalous interaction dominates the production. temperatures above EWSB instead the SU(3) JHEP04(2012)106 ], 38 (3.1) ] it was 1 ]. If the LHC 19 – is less than about ]. In ref. [ red as warm dark 17 , R 30 scussed in this paper. T 10 inos must be so light ( s are analogous to those . died in terms of cosmo- – er neutralino, the axino n in the axion models con- l candidate for CDM and 8 ss some other mechanisms ry high, above the decou- ormally not the NLSP and = 1 for the QCD anomaly at a subsequent period of is the neutralinos that play could constitute CDM [ del dependent and won’t be 3 ino population comes firstly c xinos (and that they are not a + h.c. α aY Y a C α W c ray positrons, as pointed out in Y aα aα and a α G ]. Additionally the decay of inflaton or ϑS W ϑS Y which sets 3 2 aα ]. On the other hand, when the reheating aW W 2 a d ]. d C f 37 Z 41 Z and large gluino mass, axino yield from decay keV) mass, axinos could constitute WDM. As → ϑS G , e q 2 – 11 – ∼ 40 GeV) axinos can reach thermal equilibrium with d m aY Y /N 9 a aW W a a ∼ < Z C C V 10 πf πf s 2 Y R a 8 8 α α α GeV), their relic number density reaches again thermal ∼ T > 2 2 2 πf 9 8 √ √ √ ] and collider signatures in refs. [ 10 4 and lower ( − − − – ∼ 2 R ] to include TP. If the axino mass is between around an MeV = T 2 ] and much lighter ones as hot DM in ref. [ eff (NTP) of axinos from the decay of the NLSP after it has frozen 37 L can dominate the abundance [ ˜ a q ) ]. At higher 2 → 2.9 e q GeV [ ]. The possibility of either WDM or very heavy axinos is not di 4 39 In general the couplings of the axino to gauge and matter field The interactions leading to CDM axinos were extensively stu Thermally produced axinos in the keV mass range were conside 10 1 keV) that they become warm or hot DM [ × ∼ processes 5 out from the plasma. Noteremain also in that, thermal even equilibrium, though for squarks are n non-thermal production a digression, we note that,the when role axinos of are the very LSP, heavy, their and population it from heavy axino decays < give in eq. ( logical implications in refs.does [ not confirm theCDM decay idea of with the heavy R-paritysuch squarks conservation as or can an not gluino effective be to SUSY saved a is unle introduced light [ leading in particular to theref. possibility [ of TeV-scale cosmi matter (WDM) in ref.shown [ that axinos from neutralinothis NLSP was decays extended can in beto ref. a several natura [ GeV, the correct axino CDM density is obtained whe moduli can produce axinos butconsidered such here. contributions are very mo produced directly infrom inflaton the decay), hot a thermalpling post-inflationary bath. temperature ax of If axinos ( the reheating temperature is ve SM particles andcosmic their inflation superpartners. dilutes the However, population assuming of th such primordial a Here we normalize the PQ scale by 2 equilibrium and is the same as that of photons. In that case ax sidered above. 3.1 Thermal production At sufficiently high temperatures ( coupling and therefore defines the coefficients JHEP04(2012)106 ge listed s e included, ] to include GeV. Below 6 e will discuss 6 aptures relevant 10 ontributions come ≫ R -channel exchange, are T , while it does not affect u 2 os the contributions from thod gives always positive uge-dependent in the next- results in an enhancement roximation as the only way by the Boltzmann equation ber density is much smaller . Therefore in the numerical possible effects. rks or gluinos can dominate ite-temperature propagators ng interactions we will follow ble - or y following ref. [ t 2 eff ernative method of computing and low reheating temperature proximation was applied to the o and other particles. At lower -section. Subsequently, a more ]. In fact, the production rate e definite for all values of g ts and the method how they are d physical cross-sections for each 5 in the large gauge coupling regime. squark interaction term. This term s < m , we give for simplicity only the first two 2 ], are regularized with the inclusion of an 2 1. However the logarithm in the approximate – 12 – ≫ ], an effective gluon thermal mass was introduced 2 2 eff ] to obtain the (dominant) axino TP cross-section. We ]. This procedure includes axino production via gluon 2 6 s/m 2, and therefore the result becomes unphysical. Strumia . 1 ]. In ref. [ & 4 3 , g ]. However, the HTL approximation is valid only for small gau 3 5 ]. 2 ], indicating that not all the contributions of that order ar gauge interactions were neglected, but for completeness, w gives unphysical negative value for 42 2 Y 1, which corresponds to the reheating temperature ≪ g GeV. Even though this method is also not gauge invariant, it c 4 and U(1) 10 L In the KSVZ model, at high temperatures, the most important c To start with, in evaluating the contributions from the stro As stated earlier, in the previous studies of TP of relic axin GeV the HTL approximation becomes less reliable [ . 6 R 10 infrared-divergent and thus, following ref. [ than that of photonswithout and backreaction its [ time evolution is well described temperature is below the decoupling temperature, axino num T obtained is described inleading the terms appendix. in In the table expansion for the previously ignored dimension-5 axino-gluino-squark- changes only the contribution from thethe processes other H terms. and J in The ta processes B, F,effective G gluon and thermal H, mass withdefinite in gluon values the for gluon the propagator. single cross This sections. me The full resul will further update and correct the tables presented there b to-leading order [ and therefore also does not giveFor a these fully satisfactory reasons, result we believe thatthe it rate is at worth large pursuing couplingsto an and evaluate alt the we axino apply thermal the production effective at large mass coupling app becomes even negative at decays, which is kinematicallycompared to allowed the by HTL approximation. thermal However, mass, his method and is ga the production of axinos [ to regulate the infraredconsistent calculation divergence using in the hard the thermalaxino scattering loop production (HTL) cross in ap ref. [ coupling, from two-body scatterings ofreheating colored particles temperatures, into on an the axin other hand, the decay of squa here all SM gauge groups explicitly in order to examinethe their method used previously in ref. [ physical effects like plasma screening andsingle gives scattering positive process. an SU(2) tried to improve this behaviourfor by using gluons the and full gluinos resummed in fin the loop [ calculation we have used the full formulae which are positiv in the appendix. formulae of table JHEP04(2012)106 , Y ge α (3.2) (3.3) to have a ns, which charge. represents Finally, in 7 Y D ˜ f gauge interactions. ˜ q a ) satisfy the relations Y -doublets with corre- al state and are given T L ∗ N ˜ q s (the number of initial spin g gaugino masses. s ˜ q n , and neglect Fermi blocking e at these temperatures. We i X and U(1) -sfermion-sfermion interaction is always satisfied. η a 2. and the ones proportional to L mperature of SM superpartners ˜ g / o ˜ a L 1) a , b s rt from the NLSP one, drop down to zero µν ) are the gauge coupling, the gaugino − (the number density factor, which is s α πf ( , can be written as 2 W i 4 µν b n η µν D N ˜ ¯ σ , as in the first line for SU(3), as stated f ˜ + Y gauge groups, respectively. factor, the processes A, B, and F vanish ˜ W ˜ are the model-dependent couplings for the f, a 2 a YY a ] , ] f f Y ν Y 3 s = ( T Y b µν ν ˜ 2 Y ∗ α Q D 2 G π of the gauge group SU( , γ ˜ | , ∗ , γ aY Y f b A, . . . , J 4 µ ˜ µ 2 a f g Y jk a are the sfermions carrying the U(1) C – 13 – jk γ ]˜ g γ [ = Y T α [ ν T ˜ | 5 f g 5 D n ) = ˜ , γ f jk s X ˜ aγ and ˜ and U(1) ˜ aγ f µ ( a and and X γ n P L come from the SU(2) [ ), respectively. We include above the SUSY breaking a a ˜ σ ˜ 5 a W Y 2 Y aY Y µν abc ˜ ˜ a a aW W πf α πf aW W ˜ aγ P f C a a , where C W a C 2 16 Y 2 Y n 16 , s πf πf α α σ α α and replace quark triplets with SU(2) πf 4 4 ˜ i i with the square of the corresponding hypercharges. Finally α W ), are given by 2 16 2 , + + + + | i 1) and 2 and U(1) a jk α = − 2.11 L T 11) to count the matter multiplets charged under the MSSM gau | 2 , gauge group axino-gaugino-gauge boson anomaly interactio 4 for fermions). We assume particles in thermal equilibrium eff N / 14 L ( Y , gauge groups. N ). Here Y = (the number of chiral multiplets) and = (12 2 3.1 ) denotes the cross-section averaged over spins in the initi | s F F . The relevant multiplication factors are also listed: ( abc and U(1) n 2 n N the group theory factors f σ | L 2 We start from table The total cross-sections Next we move to include contributions from the SU(2) At lower temperatures, superpartner number densities, apa 7 a,b,c below eq. ( from the U(1) SU(2) is defined after the standard normalization of where ¯ where the terms proportional to states), sponding group factors. For the abelian U(1) 1 for bosons and 3 or Bose-Einstein enhancement factors,restrict which ourselves are to close temperatures toinvolved, above on so the that the freeze-out approximation te of thermal equilibrium P table (nearly) Maxwell-Boltzmann distribution, proportional t and we can replace field and the field strength of SU(2) the sfermions of the SU(2)-doublet and The relevant axino-gaugino-gauge bosonterms, and in axino-gaugino view of eq. ( we use groups (SU(3), SU(2) in table scale the full 1-loop MSSM running of the gauge couplings and and the NTP regime is reached. JHEP04(2012)106 . s (3.4) (3.5) n be gen- . Therefore 2 eff 2 3 3 3 4 4 4 η 3 4 3 4 3 4 3 4 1 1 1 1 3 4 3 4 3 4 ]. The logarithm η 2 s < m , F F F F F F F , F 1 1 1 N ! N N account this effective n  N N N N 1 2 2 2 e ˜ 2 2 g q 2 m m ginos are heavy enough. − inematically allowed only s 4 4 2 4 2 4 4 2 4 1 n x ee-body decay is suppressed te squark masses. Therefore . The H and J entries with an 4 G ) is subdominant to the two- nel involving strong interactions. eff   − m those of ref. [ 2 2 m missing term and cross-sections or 3.3 n in the appendix. ˜ | x ∗ 1 ˜ s 2 ˜ a − 7 23 4 12 jk egative value for √ 4 7 are positive definite for all values of
T ) also generates three-body gaugino | − − − ´ ´ 2 eff π s ´ 1 + 3.3 2 eff 2 eff α )  2 eff 16 2  s/m x s/m s/m 6 s/m ` ` ! ` ) log 2 n a 2 − log ˜ g 3 f log log ∗ σ x 2  log x ˆ ˆ | 3 2 2 2 ˆ – 14 – | | | 2 2 2 m 2 2 π | | | | | 2 2 s | a a a abc jk jk jk + 2 a a a α jk jk jk a f abc T T abc jk T x | | | | 128 T T T f f T | | | | | | 2 (1 + 5 1 1 1 1 1 6 1 8 32 16 1 2 1 4 1 4 24 24 1 4

x − 4 c c c j j j j a (1 − a a ) = q q q q g g g g g g x ∗ k 1 ˜ q 4 + + ˜ + + ˜ + ˜ + ˜ + j + ˜ + + ˜ √ ˜ q − ˜ ˜ ˜ ˜ a a a a ˜ 3 ˜ 2 ˜ a a a ˜ a ˜ ˜ a a ˜ a → → → → → → → → → → → ) = ], an effective dimension-4 axino-quark-squark coupling ca b b b k k k k ∗ j j k x a 3 q q q q g g g q ( q q Process g G + + ˜ + ˜ + + + ˜ + ˜ + ¯ + ˜ + Γ(˜ a a a a a a a j k k ˜ ˜ ˜ ˜ ˜ q g g q g g g g g q I J n F E B C H A D G denotes the gluino mass, ˜ g . The cross-section for each axino thermal-production chan m The second term for each gauge group in eq. ( As stated in ref. [ in the others processes (A,B,D,E,F, and I) are corrected fro F The particle masses areasterisk neglected in except for the the thirdn plasmon column mas are changed due to including the Table 2 The full cross sections for the processes B, F, G and H are give body decay. erated at a loop level also in the KSVZ model. Here we take into decays into an axinoThe and three-body two decay rate sfermions, of assuming the that gluino is the given gau by in the approximate formulae in this table gives unphysical n where in the numerical calculation we used the full formulae which when the gluino mass is largergluino than three-body the sum decay of through the the two final-sta second term in eq. ( and the mass of the axinoby has an been additional neglected. power However, the of thr the gauge coupling constant and is k JHEP04(2012)106 ) ), 3.7 evel (3.7) (3.9) (3.8) (3.6) ] and 2.27 4 ]; com- 58 ng contained in ] for the effective 10 ominantly from the – 8 rection of the order of r row to the down-type recently investigated also rk, the tree-level coupling k-squark coupling which is iggsino coupling [ rk decays through eq. ( rst obtained in ref. [ interaction term in eq. ( β β, KSVZ models and a tree level mportant for the non-thermal . the one-loop ones for the light 2 2 ow the EWSB scale. Similarly, 2 tering channels contributing to loop term and is proportional is important for axino thermal c ˜ ightest colored superpartner and a,  5 ˜ cos sin g a γ f m ( q  a R/L f , m P j i a is larger than the top quark mass. The log ¯ f ψ ∓ µ ˜ g 2 a µ 4 f m = √ J/R j ], 2 ˜ ψ 3 2 π = 2 s 2 – 15 – d,u v 2 α v ], coming from the eff L/R eff √ q a g g ] in a full two-loop computation. This coupling is 43 f m i = 10 ≃ ∓ – ∓ . It contributes to axino TP through higgsino decays in ˜ ψ 8 a ), = V ˜ aψ L/R eff 1 for the same choice of the gluino mass. Note that, at low L g 2.13 = 2 ∼ > L/R eff a g β f 4 for the gluino mass of 700 GeV, while the bottom quark tree-l and ∼ P < denote the SM fermions and their superpartners. β /M j 2 ˜ a ψ V ), or even become much larger if s f ]. Note that the Yukawa coupling and the axino-higgsino mixi and 3.8 ∼ ], j 43 3 µ ψ [ 30%, in analogy with what has recently been obtained in ref. [ ˜ g In the DFSZ models, there is also a tree-level axino-Higgs-h In the DFSZ models there exists also a tree-level axino-quar In the KSVZ class of models, the effective coupling comes pred − m Subleading terms have not20 yet been computed and may give a cor where axino production due to thisin coupling ref. in [ DFSZ models has been thermal equilibrium, and can be comparable to that from squa coupling dominates if tan (with a tiny mixing) in the DFSZ models [ Yukawa interaction, which appears at a two-loop level in the generations, but not fordominates the if third tan one. In fact for the top qua where the upper rowquarks. relates These to tree-level the couplings up-type are always quarks smaller and than the lowe remains in equilibrium to ratherthere low temperatures, exists even an bel effective tau-stau-axino vertex, which was fi reheating temperatures, only theproduction. top-stop-axino This coupling is because the lighter stop is usually the l and eq. ( the neutralino mass matrix also give rise to additional scat pare the second term in eq. ( more recently re-derived insmaller and ref. not [ important forproduction thermal when production, the but stau it is is the i NLSP. tau-stau-axino coupling. proportional to the mass of the quark [ logarithmically divergent partto of the gluon-gluino-quark where JHEP04(2012)106 ) for the ], which m strong 2 3.7 from strong R = 1 TeV. We T ], the inclusion ˜ g 2 m = ) and eq. ( 1010101010101010 e q ]. For different values 3.3 1010101010101010 3 m 1111111111111111 ,,,,,,,, r comparison, the axino 99999999 lly and present the results , we show the axino yield figure 2 of ref. [ ture, while the other new f. [ 1 1010101010101010 lot in ref. [ ffective mass approximation 88888888 is a factor 3 difference in the h dimension-4 scatterings are re in the instantaneous reheating 1010101010101010 77777777 GeV and ey). We also denote the yield from 1010101010101010 ] is plotted with the blue (dashed)

d), as well as out-of-equilibrium bino- 11

actions in eq. (

). We use the representative values of 5

HTL approx. approx. approx. approx. approx. approx. approx. approx. HTL HTL HTL HTL HTL HTL HTL HTL Strumia Strumia Strumia Strumia Strumia Strumia Strumia Strumia 66666666 . In figure for U(1)for U(1)for U(1)for U(1)for U(1)for U(1)for U(1)for U(1) 2 a 1010101010101010 f = 10 55555555 squark decaysquark decaysquark decaysquark decaysquark decaysquark decaysquark decaysquark decay gluino decaygluino decaygluino decaygluino decaygluino decaygluino decaygluino decaygluino decay neutralino decayneutralino decayneutralino decayneutralino decayneutralino decayneutralino decayneutralino decayneutralino decay a 1010101010101010 f (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) RRRRRRRR 44444444 TTTTTTTT scatteringscatteringscatteringscatteringscatteringscatteringscatteringscattering 1010101010101010

– 16 – Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,

]. eff. mass approx. approx. approx. approx. approx. approx. approx. mass mass mass mass mass mass mass eff. eff. eff. eff. eff. eff. eff. eff. mass approx. mass eff. 33333333 3 1010101010101010 GeV, SU(3) only GeV, SU(3) only GeV, SU(3) only GeV, SU(3) only GeV, SU(3) only GeV, SU(3) only GeV, SU(3) only GeV, SU(3) only Axino Thermal Production Production Production Production Production Production Production Production Axino ThermalAxino ThermalAxino ThermalAxino ThermalAxino ThermalAxino ThermalAxino ThermalAxino Thermal 22222222 1111111111111111 as a function of the reheating temperature 1010101010101010 =10=10=10=10=10=10=10=10 aaaaaaaa 11111111 ffffffff TP ˜ a 1010101010101010 Y 00000000 = 1 TeV. For comparison, we also show the HTL approximation ] is shown with the green line. ˜ g 1010101010101010 00000000 6 -4-4-4-4-4-4-4-4 -5-5-5-5-5-5-5-5 -6-6-6-6-6-6-6-6 -7-7-7-7-7-7-7-7 -8-8-8-8-8-8-8-8 -9-9-9-9-9-9-9-9 -1-1-1-1-1-1-1-1 -2-2-2-2-2-2-2-2 -3-3-3-3-3-3-3-3 m -14-14-14-14-14-14-14-14 -15-15-15-15-15-15-15-15 -10-10-10-10-10-10-10-10 -11-11-11-11-11-11-11-11 -12-12-12-12-12-12-12-12 -13-13-13-13-13-13-13-13 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010

1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010

=

axino axino axino axino axino axino axino axino

Y Y Y Y Y Y Y

Y e q

TP TP TP TP TP TP TP TP is the ratio of the number density to the entropy density) fro for representative values of m 2 n/s ≡ and Y 1 GeV and . Thermal axino yield 11 the curves move up or down proportional to We have evaluated the thermal production of axinos numerica a = 10 (where f a interactions using the effectivef mass approximation (black interaction in the KSVZ model.is Our shown result with obtained the with the solid e black line. Compared to the previous p (dotted blue/dark grey) andsquark (solid that green/light of grey) Strumialike and neutralino (green/light decay gluino gr (dashed decay black). (dotted Here re we used the inter Figure 1 squark interactions do not haveabundance any at noticeable high effect. reheating There was temperature a compared numerical to erroryield that at from in that scatterings using timeline the and and HTL was Strumia’s approximation corrected result [ [ later. Fo of Y of the squark decay changes the plot at low reheating tempera the axino production, but weusually will less neglect important them than here the since suc decays [ in figures KSVZ model. We useapproximation the for same the definition three of methods. reheating temperatu do not consider here the dependence on masses, however see re JHEP04(2012)106 on in- known nteractions only, for 101010 squarks and gluinos in 101010 111111 ,,, ld require a gluon ther- 999 re, which in our scheme 101010 difference between the two ], and this is confirmed in cays do not give significant me that this error does not e contribution from an out- 3 888 [ ould reabsorb this difference scattering diagrams involving ical error of using the effective 101010 e could match the perturbative those coming from dimension-5 R ongly and in the opposite direction 777 T 101010 666 ], start playing some role. We mark from each of the SM gauge groups. Here, 101010 6 GeV and we apply the effective mass are not perfectly parallel due to the running R 4 555 T R 101010 (GeV) (GeV) (GeV) T

used in our calculations. Hence we prefer

RRR

SU(2) SU(2) SU(2) U(1) U(1) U(1) SU(3) SU(3) SU(3) 444 TTT 101010 gT Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi, – 17 – 333 ∼ 101010 GeV, SU(3), SU(2), U(1)U(1)U(1) GeV, SU(3), SU(2), GeV, SU(3), SU(2), GeV, SU(3), SU(2), 222 111111 101010 =10=10=10 as a function of aaa 111 fff with eff. mass approx.with eff. mass approx.with eff. mass approx. 101010 TP ˜ a 000 Y 101010 000 = 1. The lines at high -4-4-4 -5-5-5 -6-6-6 -7-7-7 -8-8-8 -9-9-9 -1-1-1 -2-2-2 -3-3-3 -14-14-14 -15-15-15 -10-10-10 -11-11-11 -12-12-12 -13-13-13 101010

101010 101010 101010 101010 101010 101010 101010 101010 101010

101010 101010 101010 101010 101010 101010

axino axino axino aY Y Y Y Y

TP TP TP C = GeV the axino abundance using the effective mass approximati with a green solid and red dashed curves, respectively. It is 4 1 10 aW W & C R . Thermal axino yield T . While all the above contributions are generated by strong i 1 For lower temperatures the contributions from the decays of For increase for reheating temperatures less than 10 instead to consider this factormass as approximation an at estimate high of reheating the temperatures. theoret We assu approximation to the DFSZ model. thermal plasma, which were not included in ref. [ Figure 2 comparison, we show also (as a black dashed line) the relativ that, at reheating temperaturesdimension-4 above operators superpartner are masses usually subdominantoperators relative and to to decaycontribution terms, to and the are negligible. TPfigure of Also the axinos, de apart from very low those in figure we have used of the gauge couplings,than which the affects other the gauge SU(3) groups. yield more str in the definitioncannot of be the determined effective self-consistently. With gluonresult this mass at tuning high at w temperature. highmal temperatu mass On smaller the than other the hand, expression doing this wou creases consistently with thatprescriptions of is Strumia. of We order found a that factor the of three. In principle we c JHEP04(2012)106 ], 44 = 0 and L em Q = 1 as a aY Y ion from each C ) unification [ , e c d 101010 , well below squark or tude estimate of these 101010 111111 ,,, R = 1 and 999 ormalized value in figure T ay bring a modification of 101010 =8/3)=8/3)=8/3) ) are significantly suppressed versally, while the goldstino photon originally considered 888 =0)=0)=0) nt from the case of gravitino aW W aYYaYYaYY al to a gauge coupling-squared 101010 C aYYaYYaYY = 200 GeV. The horizontal lines m scatterings due to SU(2) ree gauge groups are of the same 777 no abundance gives the correct DM that these channels could become ˜ h 101010 m =2/3 (C=2/3 (C=2/3 (C =0 (C=0 (C=0 (C 666 for two specific KSVZ models: ,e) unif,e) unif,e) unif aY Y ememem ememem ccc 101010 R T ,C 555 101010 (GeV) (GeV) (GeV) the curves move up and down. We note RRR 444 TTT DFSZ (dDFSZ (dDFSZ (d KSVZ QKSVZ QKSVZ Q KSVZ QKSVZ QKSVZ Q aW W 101010 C Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi, – 18 – aY Y 333 GeV GeV GeV C -6-6-6 101010 GeV GeV GeV 222 111111 =10=10=10 3), and for a DFSZ model with the ( / 101010 . Thus it would be only for very large (and perhaps and

3 as a function of =10=10=10 -4-4-4 -2-2-2 aaa 111 axinoaxinoaxino α fff = 8 1 1 1 100100100 mmm 101010 101010 101010 TP ˜ a ∝ Y 000 aW W (red dashed), by a factor of 10 or more. This is because the aY Y σ 101010 C 000 c C -4-4-4 -5-5-5 -6-6-6 -7-7-7 -8-8-8 -9-9-9 -1-1-1 -2-2-2 -3-3-3 -14-14-14 -15-15-15 -10-10-10 -11-11-11 -12-12-12 -13-13-13 101010 101010 101010 101010 101010 101010 101010 101010 101010 101010 3 (

101010 101010 101010 101010 101010 101010

/

axino axino axino Y Y Y

TP TP TP = 200 GeV and the higgsino mass = 2 µ em Q , we show a contribution to the axino yield in thermal product 2 . Thermal axino yield ]. It is clear that NTP is only important at very low 1 couplings (blue dotted and green solid lines, respectively = 0) and Y In figure aY Y C Figure 3 SM gauge group interaction. Here we setU(1) the coefficients for which we used compared to that ofinteraction SU(3) between axinos and gaugeso bosons are that proportion the cross-section is normalization. As shown in the figure, the contributions fro relic density. of-equilibrium bino-like neutralino decayin to ref. an [ axino and a show the values of axino mass for which the corresponding axi unnatural) values of the effective couplings ( gluino masses. 2. For different values of comparable to the QCDeffects, contribution. we included To the give SU(2) an and order U(1) of contributions magni with a n that in general SUSYthe breaking couplings effects here in considered. theproduction since leptonic The the sector interactions situation m of hereorder: the gravitino is to the differe the spin-3/2 th gravitino component couples in fact uni JHEP04(2012)106 nce up- . ˜ a Y ant than the and typically aY Y C on is played by the he DFSZ model was anomaly term starts L ss as small as 100 keV, mmetry breaking is the VSZ and DFSZ type of , axino production from 1000 GeV, axino produc- vy PQ charged states can µ for reheating temperature her hand the neutralinos, scription in the scattering − ings. Therefore the use of re conclude that the QCD el in the range of reheating l lines corresponding to the he axino production at high trum, can freeze-out with a elic abundance of er colored particles becomes = 200 GeV and the higgsino tional to ome important. Actually, the 100 µ ∼ mal axino production, as we have < has been recently called “freeze-in” in , the SU(2) as in the KVSZ case, but with a R R T T R T the contribution from neutralino decay ]. In this range of reheating temperature, 58 aY Y 9 C 10 GeV and it is very suppressed. Moreover we – 19 – ∼ ]. 3 R ]. The freeze-in mechanism was included already in gluino or T couplings becomes dominant in the very low reheat- 46 . a Y f There we show the KSVZ model with different values of 8 . 3 the thermal production from scatterings becomes strongly s ] and our result is consistent with that analysis. The abunda R 43 T . This contribution to the axino yield is usually more import 1 . coupling. In the case of non-zero 12 = 200 GeV. We can see that even for relatively small GeV. The importance of axino-Higgsino-Higgss coupling in t − 6 ˜ h ] and discussed for the axino in ref. [ 10 m aY Y Our results for the total thermal production yield for both K For the case of the DFSZ instead, the role of the QCD interacti However, at low C 45 A similar figure for the DFSZ model is given in ref. [ Such general effect due to decaying particles in equilibrium is practically model independent as long as the number of hea . 8 9 R R SU(2) interaction and thehiggsino dominant one decay instead term of above the the squark EW one. sy models can be seen in figure below 10 is so large thatindependently the CDM of density the can be reheating reached temperature. with an axino ma one due to neutralino decays in equilibrium, which is propor give also the yield for the DFSZ model in solid green, for still substantial number and then giveseen rise also in to figure non-ther ing temperature regime sincestrongly the number suppressed density by of the thedepending Boltzmann heavi on factor their there. composition and On the the supersymmetric ot spec mass contribution is strongly dominant in theT KVSZ models and so t component proportionally to the gaugino masses. We therefo in equilibrium can be clearly seen for be absorbed into the definition of pressed by the Boltzmann factor. In the region where ref. [ squark decays to axinos in the plasma in ref. [ process is irrelevant to thetemperature axino where production the in decay the dominates. DFSZ For mod higher the axino production fromthe decay effective dominates mass that approximation from or scatter another IR screening pre the higgsino decay dominates over theT one from the anomaly terms tion due to the decayslightest of neutralino gaugino, decay squarks or via neutralinos U(1) bec recently discussed in [ dominating and the abundance is proportional to smaller coefficient. In theaxino same mass figure, giving we the also correct mark DM horizonta relic density for the given r JHEP04(2012)106 (3.10) (III). The upper 8 ponent, we need − . 222222 rmal plasma. For the 101010101010 NLSP 111111111111 ,,,,,, Y P decays after theyr have reheating temperatures 111111 (I)(I)(I)(I)(I)(I)  erature is high enough for (II)(II)(II)(II)(II)(II) 101010101010 (II), and 10 axinos, the NLSP must itself ) ) ) ) ) ) 000000 ˜ a nihilation and co-annihilation 10 alino co-annihilation processes. ty from NTP is simply given by DMDMDMDMDMDM − 101010101010 and NTP included. To parametrize m TPTPTPTPTPTP undance of axinos. The regions disal- tons. For the stau NLSP, the yield 100 GeV -1-1-1-1-1-1 ΩΩΩΩΩΩ 30 MeV, with a neutralino NLSP). ashed lines and arrows for, respectively, (III)(III)(III)(III)(III)(III)  101010101010 > > > > > > ∼ < 10 ˜ a -2-2-2-2-2-2 axinoaxinoaxinoaxinoaxinoaxino m ]. In this case, the axino abundance is 10 = 0 (I), 10 101010101010 2 TPTPTPTPTPTP (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) × , ΩΩΩΩΩΩ -3-3-3-3-3-3 EXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDED (((((( 1 7 . axinoaxinoaxinoaxinoaxinoaxino 101010101010 NLSP too warmtoo warmtoo warmtoo warmtoo warmtoo warm 2 mmmmmm Y Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi, – 20 – -4-4-4-4-4-4 ≃ 101010101010 neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP) GeV in the KSVZ models. The bands inside like curves )) and NTP ( -5-5-5-5-5-5 (NTP from (NTP from (NTP from (NTP from (NTP from (NTP from GeV GeV GeV GeV GeV GeV NLSP 11 101010101010 111111111111 4.2 Ω -6-6-6-6-6-6 =10=10=10=10=10=10 = 10 ˜ a aaaaaa (TP)(TP)(TP)(TP)(TP)(TP) 101010101010 ffffff too warmtoo warmtoo warmtoo warmtoo warmtoo warm a NLSP m f -7-7-7-7-7-7 m 101010101010 000000 666666 555555 444444 333333 222222 111111 999999 888888 777777 for

=

101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010

R R R R R R ˜ a (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) T T T T T T m ˜ NTP a Ω versus R T 5keV, see text below eq. ( . ∼ < ˜ a m To see if such production is sufficient to give a dominant DM com to know the yieldneutralino of NLSP NLSPs yield, afterwith relevant they the processes charginos, have next-to-lightest frozen include neutralinosis out and pair-an determined slep of by the the the stau-stau annihilation and stau-neutr Clearly, in order to producehave a an substantial energy NTP density population larger of than the present DM density. correspond to a correct relicthe density non-thermal of production DM axino of with axinos both we TP used lowed by structure formation are markedTP with vertical ( blue d right-hand area of the plot is excluded because of the overab Figure 4 As stated above, axinosfrozen can out be of produced equilibrium.below non-thermally This in the NTP NLS mass mechanismindependent of is of the dominant the fo gluino reheatingthe and temperature NLSP squarks to as thermalize [ long before as freeze-out. the Axino relic temp densi 3.2 Non-thermal production JHEP04(2012)106 . 3 GeV. 9 10 222222 222222 × 101010101010 101010101010 111111111111 111111111111 ,,,,,, ,,,,,, 111111 111111 = 5 (I)(I)(I)(I)(I)(I) (I)(I)(I)(I)(I)(I) (II)(II)(II)(II)(II)(II) a 101010101010 101010101010 (II)(II)(II)(II)(II)(II) f ) ) ) ) ) ) ) ) ) ) ) ) 000000 000000 GeV GeV GeV GeV GeV GeV DMDMDMDMDMDM DMDMDMDMDMDM 101010101010 101010101010 111111111111 TPTPTPTPTPTP TPTPTPTPTPTP -1-1-1-1-1-1 -1-1-1-1-1-1 ΩΩΩΩΩΩ ΩΩΩΩΩΩ =10=10=10=10=10=10 (III)(III)(III)(III)(III)(III) (III)(III)(III)(III)(III)(III) aaaaaa 101010101010 101010101010 ffffff > > > > > > > > > > > > -2-2-2-2-2-2 -2-2-2-2-2-2 axinoaxinoaxinoaxinoaxinoaxino axinoaxinoaxinoaxinoaxinoaxino 101010101010 101010101010 TPTPTPTPTPTP TPTPTPTPTPTP (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) ΩΩΩΩΩΩ ΩΩΩΩΩΩ -3-3-3-3-3-3 -3-3-3-3-3-3 EXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDED (((((( EXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDED (((((( axinoaxinoaxinoaxinoaxinoaxino axinoaxinoaxinoaxinoaxinoaxino 101010101010 101010101010 mmmmmm mmmmmm Choi, Covi, Kim, Roszkowski Choi, Covi, Kim, Roszkowski Choi, Covi, Kim, Roszkowski Choi, Covi, Kim, Roszkowski Choi, Covi, Kim, Roszkowski Choi, Covi, Kim, Roszkowski Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi, – 21 – but with DFSZ model used in figure -4-4-4-4-4-4 -4-4-4-4-4-4 but with the PQ scale 4 101010101010 101010101010 4 too warmtoo warmtoo warmtoo warmtoo warmtoo warm neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP) GeV GeV GeV GeV GeV GeV -5-5-5-5-5-5 -5-5-5-5-5-5 (NTP from (NTP from (NTP from (NTP from (NTP from (NTP from 999999 101010101010 101010101010 too warmtoo warmtoo warmtoo warmtoo warmtoo warm -6-6-6-6-6-6 -6-6-6-6-6-6 neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP)neutralino NLSP) =5x10=5x10=5x10=5x10=5x10=5x10 (NTP from (NTP from (NTP from (NTP from (NTP from (NTP from aaaaaa (TP)(TP)(TP)(TP)(TP)(TP) (TP)(TP)(TP)(TP)(TP)(TP) 101010101010 101010101010 ffffff too warmtoo warmtoo warmtoo warmtoo warmtoo warm too warmtoo warmtoo warmtoo warmtoo warmtoo warm -7-7-7-7-7-7 -7-7-7-7-7-7 101010101010 101010101010 000000 666666 555555 444444 333333 222222 111111 000000 999999 888888 777777 666666 555555 444444 333333 222222 111111 999999 888888 777777

101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010

R R R R R R R R R R R R (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) T T T T T T (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) T T T T T T . The same as figure . The same as figure Figure 6 Figure 5 JHEP04(2012)106 ), the (3.11) (3.12) 3.6 GeV. 9 ir number density . 10 a 222222 ined Minimal Super- f tau NLSP, which can × 101010101010 eutralino if it is lighter 111111111111 ,,,,,, o an axino and a photon pe given in eq. ( GeV, for which even the 111111 ufficient axino abundance = 5 (I)(I)(I)(I)(I)(I) a 101010101010 (II)(II)(II)(II)(II)(II) 12 , f 9 TeV, which may thermalize ) ) ) ) ) ) . 10  roduce any substantial axino 000000 , au NLSP, including four-body DMDMDMDMDMDM  101010101010 < TPTPTPTPTPTP χ a -1-1-1-1-1-1 ΩΩΩΩΩΩ f ˜ m τ (III)(III)(III)(III)(III)(III) 101010101010 > > > > > > m 100 GeV -2-2-2-2-2-2  100 GeV axinoaxinoaxinoaxinoaxinoaxino ], for 101010101010 12 TPTPTPTPTPTP (GeV) (GeV) (GeV) (GeV) (GeV) (GeV)  4 − ΩΩΩΩΩΩ -3-3-3-3-3-3 12 EXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDED (((((( − 10 axinoaxinoaxinoaxinoaxinoaxino 101010101010 ] also for larger values of mmmmmm Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi, – 22 – 10 × 10 -4-4-4-4-4-4 but with the PQ scale – × 101010101010 6 8 10) 01 GeV GeV GeV GeV GeV GeV -5-5-5-5-5-5 . 999999 ∼ 0 101010101010 ]. too warm(NTP)too warm(NTP)too warm(NTP)too warm(NTP)too warm(NTP)too warm(NTP) (1 ≃ -6-6-6-6-6-6 48 =5x10=5x10=5x10=5x10=5x10=5x10 ˜ τ ≃ too warm (TP)too warm (TP)too warm (TP)too warm (TP)too warm (TP)too warm (TP) aaaaaa , 101010101010 ffffff Y χ 47 Y -7-7-7-7-7-7 101010101010 000000 666666 555555 444444 333333 222222 111111 999999 888888 777777

101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010 101010101010

R R R R R R (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) T T T T T T . The same as figure Figure 7 8 TeV) usually remain in thermal equilibrium so long that the . -boson, the NTP production is usually more efficient. For the s Z In conclusion, for neutralino NLSP, which decays mainly int These two choices for the NLSP were considered in the Constra Regarding other NLSPs, colored relics (or even a wino-like n stau lifetime is ofhadronic order decays, 1s, was or discussed less. in ref. Recently, [ the case of st or a decay to an axino and a tau-lepton through a coupling of the ty symmetric Standard Model (CMSSM) in ref. [ for the stau. Noteto that explain in the the whole latter DM case densityonly only the at for NTP accordingly stau can masses higher produce above temperatures. 1 s for a bino-dominated neutralino, and A typical relic abundance is contribution is smaller, but can still be substantial. than 1 population after freeze-out [ after freeze-out becomes negligible and therefore cannot p JHEP04(2012)106 ] ]. 49 58 (4.1) ]. ecays can He severely 10 4 – 8 tion in the DM 35 in the larger . 0 ] and reionization ]). However for the ameter space above (red dash), denoted 55 < [ 54 8 α − DM d state with Ω ntributions of axinos, the rger [ milar figures in the DFSZ / role of the upper bound on s typically around 1 sec, or er, for longer lifetimes such nos, in the range given by man- .g. the axion. ] (although in specific cases GeV, respectively. We have ight elements. For axino DM, sed at scales below their free- 9 53 ] leads to an upper limit on the rticles during or after Big Bang , on the other hand, non-trivial WDM a d neutralino yield after freezout 10 55 f y discussed in ref. [ ( keV), thermally produced axinos × . sec [ O 3 range derived from WMAP-7 data [ 113 data [ 10 . σ 0 α GeV, the BBN constraint can be avoided × < 12 (green solid) and 10 2 h 10 10 ˜ a GeV and 5 − < Ω , where we present the reheating temperature – 23 – 11 5 a < f , which can be compared to the figures in ref. [ s for pure WDM, or otherwise in the case of mixed 7 / ]. This leads to an upper bound on the decay products = 10 and 109 . a 52 0 4 f – and 6 50 013 km . 0 < = 0 (black solid) 10 WDM i v NLSP h Y ]. However, the non-thermal population of axinos from NLSP d 2 ] give a bound on the velocity of the WDM component and its frac ), corresponds to the thin bands between like curves. The par 56 4.1 This can be seen in figures constraints arise, especially for the stau NLSP, as recentl due to the small lifetime of the NLSP. For larger values of 4.1 The relic densityFor the of total axino dark DM matter relic density, we apply the 3 4 Cosmological constraints constrains its lifetime to be less than roughly 5 data [ density. A recent analysis using theaverage SDSS velocity, Lyman- 4.3 Structure formation The density perturbation due tostreaming axino length. population When is the suppres axinobecome mass cold is larger [ than still have a large velocity dispersion and can be too warm. Ly with gravitino as DM, this canparameters be considered up in to our an study, order i.e. of magnitude la Nucleosynthesis (BBN) epoch may disrupt the abundancesthe of lifetime l of theless, NLSP, and such therefore as constraints the fromconstraints neutralino become the important or BBN [ the are stau, weak.of the i Howev NLSP for a given NLSP lifetime. For the stau NLSP, a boun model are presented in figures 4.2 Nucleosynthesis An injection of high energy electromagnetic and hadronic pa the curves is excluded as giving too much relic abundance. Si versus the axino mass plane for cold/warm DM the WDM fraction is constrained to be Ω included both the thermallatter assuming and the non-thermal production co This produces a stripe inthe the relic parameter density space when and there also are plays additional a DM components, e also as (I), (II),will and (III), lie respectively. between Aeq. (I) typical ( and stau an (III). A correct relic density of axi velocity region. JHEP04(2012)106 ] , 6 8 , 5 (4.3) (4.2) (4.4) instead panned 9 GeV and as- 11 , ], especially [ on the axino mass n figure 2  ] For stau NLSP with = 10 , 1 57 ed by the overdensity a f a GeV . 5 MeV, respectively. We . y [ f lly can be obtained from  GeV, respectively. These 11 uction, taking into account 9 2 tudy [ 10 / m the free-streaming velocity 10 1 inos produced through NTP is a  GeV ]. For the bino-like neutralino LSP of 100 GeV, as in figure  2 f × . / 11 56 1 dependence arises leading to non-  − , 10 ˜ τ ˜ a 4  ˜ a  m , m 1 m we have fixed 2 − 5 keV for TP axinos. This lower bound 150 MeV and 7 1 keV 100 GeV χ 8  ), NTP is always subdominant so that the   & & m 1 s ˜ a ˜ a GeV and 5 / − 100 GeV χ 3.12 m m  11 remain vertical and practically independent of  m – 24 – 1 ˜ a R − 100 GeV T In figure m  065 km = 10  . 1 MeV 0 a with a vertical blue dashed line and an arrow. For the ˜ a × 10 ]. f  5 ≃ s m 10 0 / – 1 MeV v 8 100 GeV), which is a typical value for bino-like neutralino data implies  / and s α 2km with a vertical blue dashed line and an arrow. / 4 , as given in eq. ( 5 ≃ 13 NLSP 0 − 4km 100 GeV) has been assumed, typical of stau NLSP. So long as the v m . / 3, we find ( and 0 10 5 MeV when , we show contours of the reheating temperature in the plane s / . 9 4 12 30%). ∼ ≃ < − 0 NLSP = 8 − Y v m and ( 20 = 10 8 aYY 14 ∼ > C − NLSP Y 30 MeV and 1 = 10 The free-streaming velocity of axinos produced non-therma The present velocity of thermally produced axinos is given b For NTP axinos and for the neutralino NLSP we find a lower limit In figures & Similar figures are shown in ref. [ , while as soon as the NTP becomes important, ˜ ˜ a a 10 NLSP vertical curves. For example, with a bino-like neutralino N m and for the stau NLSP, sumed TP dominates, the curves of constant Therefore the above Lyman- 5 Conclusion We have performed an updatedsome analysis new of the calculations relic that axino have prod appeared after the initial s the NTP contributes only upsmall abundance to 10 % ofcontours the remain axino vertical. LSP In DMof this density. the case NTP the axinos bounds are coming absent. fro Y stau NLSP the analogous lower bounds are the lifetime of theNLSP NLSP with and the mass relations [ NLSP. The cyan wedgeof in DM, the while upper in right-hand the corner red is wedge exclud below it the axino is not the LSP. I is marked in figures m stress that these bounds applysubstantial solely ( if the population of ax limits are marked in figures by the NLSP and the axino mass. JHEP04(2012)106 and GeV. 11 aW W , which do C = 10 /s a f 2 eff m in 333333 fter the latest calcula- nates around the weak 101010101010 timistic: in the KSVZ- 111111111111 ,,,,,, arge and the convergence o explore the question of al production mechanism, king it independent of the ino mass plane. Here we have onal anomaly couplings is article decay contributions rent results, we estimate the 222222 upling. ly to be still of order a factor ee from the uncertainty in the 101010101010 he overdensity of DM, while in the s of the coefficients ies above both estimates given in uring their positivity in the whole 111111 101010101010 (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) 200200200200200200 100 100 100 100 100 100 70 70 70 70 70 70 000000 axinoaxinoaxinoaxinoaxinoaxino mmmmmm Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi, 101010101010 – 25 – -1-1-1-1-1-1 GeV GeV GeV GeV GeV GeV 111111111111 101010101010 GeV GeV GeV GeV GeV GeV ======333333 RRRRRR =10=10=10=10=10=10 100 GeV), typical of neutralino NLSP, and taken aaaaaa TTTTTT 101010101010 / ffffff -2-2-2-2-2-2 101010101010 NLSP 222222 333333 GeV, and to that one has to add also some possible (unknown)

4 m

101010101010 101010101010

NLSP NLSP NLSP NLSP NLSP NLSP ( (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) m m m m m m 10 12 − ∼ R T = 10 NLSP ], and this seems natural since we included subleading terms . Contours of constant reheating temperature in the NLSP-ax Y 6 , ]. This is probably not surprising since the QCD coupling is l 5 . Instead for DFSZ-type models, the Yukawa interaction domi 6 We have found that the uncertainty has not really decreased a Regarding the model dependence, our conclusions are more op aY Y of the perturbative seriesuncertainty is in quite the slow. relicof density Comparing 10 of the or axinos diffe produced so at thermal contributions due to non-perturbative effects.refs. Our [ result l assumed tion [ for the thermal part,uncertainty and and compared model-dependence. them with our own results t The cyan wedge in thered upper wedge right-hand below corner it is the excluded axino by is t not the LSP. Figure 8 start playing thecompletely dominant negligible, role. apart from The unnaturally inclusion large of value the additi indeed increase the cross-sections forrange single of channels integration, ens even in the limit of verytype large the gauge QCD co anomaly termapart strongly dominates from the the axino therm case of small reheating temperatures where sp C scale and can give thereheating main temperature. production mechanism Therefore of the axinos axino ma abundance is fr JHEP04(2012)106 of these PI 1 C authors claim a ] appeared which GeV). 58 11 cay can produce the 333333 10 Yukawa couplings less nterestingly enough, a 101010101010 ently large abundance. 111111111111 ,,,,,, ≃ e. a is ref. [ re has to be very low and f ype of Feynman graph and ost cases the gluon thermal t is the EW anomaly term n dominate, unless the stau coupling in the relevant dia- h for NLSP thermalization). 100 GeV), typical of stau NLSP. 222222 e the whole DM density at the ntum is much larger than the / momentum dependence in the 101010101010 PI 1 C NLSP GeV for m uplings. The coefficient 6 ( 111111 14 101010101010 − (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) = 10 000000 axinoaxinoaxinoaxinoaxinoaxino mmmmmm Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi, 101010101010 – 26 – NLSP Y -1-1-1-1-1-1 GeV GeV GeV GeV GeV GeV 111111111111 101010101010 ======RRRRRR =10=10=10=10=10=10 aaaaaa TTTTTT 200 GeV200 GeV200 GeV200 GeV200 GeV200 GeV 100 100 100 100 100 100 70 70 70 70 70 70 ffffff of the axino production from the dimension-5 operators for except for 8 2 -2-2-2-2-2-2 101010101010 222222 333333 /T

2

101010101010 101010101010

NLSP NLSP NLSP NLSP NLSP NLSP

(GeV) (GeV) (GeV) (GeV) (GeV) (GeV) m m m m m m M , i.e. for the heavy quark mass less than 10 of the PQ-charged fermions in the loop. Due to this effect, the 5 . The same as figure − M During the final stages of completion of this work, the analys For both models, also the non-thermal production via NLSP de We investigated such suppression by inserting their suppression of order the DFSZ case andthan for 10 extremely small KSVZ quark masses (with Figure 9 discusses in details axinoone-particle couplings irreducible one-loop and axino-gluon-gluino finds co ainteractions non-trivial is suppressed whenmass the external particle mome the axino and NLSPlight bino masses NLSP not decaying out toocost of hierarchical. equilibrium of can a still We very produc findFor low the that, reheating stau i temperature case (but instead,NLSP sufficiently it yield hig is after quite freeze-out unlikely is that unusually the large. NTP ca method used for thethat IR-divergence. dominates, giving At a lower large abundance temperatures, than i in therequired KSVZ axino cas DM density,But if for the this mechanism NLSP to decouples dominate, with the a reheating suffici temperatu grams and we obtain differenta suppressions strong depending dependence on on the the t IR regulator contained there, in m JHEP04(2012)106 ] for 58 ] and our 58 DM axino, on in ref. [ Cold GeV. For the KVSZ 22222222 o production is dom- PI 7 1 1010101010101010 − 1111111111111111 ,,,,,,,, C 6 pendent Junior Research 11111111 (I)(I)(I)(I)(I)(I)(I)(I) nt funded from the Korean YC acknowledges the Max (II)(II)(II)(II)(II)(II)(II)(II) 1010101010101010 TP). LC would like to thank investigation of the two-loop nce and Technology (MEST), GeV GeV GeV GeV GeV GeV GeV GeV ue. . For large fermion masses in 00000000 1111111111111111 GeV. The 1010101010101010 ) ) ) ) ) ) ) ) e and its Program “Dark Matter: 5 -channel gluon exchange for van- t ality when this work was initiated. =10=10=10=10=10=10=10=10 coupling and a gluon thermal mass DMDMDMDMDMDMDMDM aaaaaaaa -1-1-1-1-1-1-1-1 ffffffff (III)(III)(III)(III)(III)(III)(III)(III) TPTPTPTPTPTPTPTP 1010101010101010 = 10 PI ΩΩΩΩΩΩΩΩ 1 -2-2-2-2-2-2-2-2 Q > > > > > > > > C aaaaaaaa 1010101010101010 m TPTPTPTPTPTPTPTP (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) ΩΩΩΩΩΩΩΩ -3-3-3-3-3-3-3-3 EXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDEDEXCLUDED (((((((( axinoaxinoaxinoaxinoaxinoaxinoaxinoaxino 1010101010101010 too warmtoo warmtoo warmtoo warmtoo warmtoo warmtoo warmtoo warm mmmmmmmm Choi, Covi, Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Kim, Roszkowski Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi,Choi, Covi, – 27 – -4-4-4-4-4-4-4-4 1010101010101010 GeV and Neutralino NLSP)Neutralino NLSP)Neutralino NLSP)Neutralino NLSP)Neutralino NLSP)Neutralino NLSP)Neutralino NLSP)Neutralino NLSP) 6 -5-5-5-5-5-5-5-5 (NTP from (NTP from (NTP from (NTP from (NTP from (NTP from (NTP from (NTP from GeV GeV GeV GeV GeV GeV GeV GeV 55555555 1010101010101010 but showing as well in magenta the yield suppression found in = 10 -6-6-6-6-6-6-6-6 GeV GeV GeV GeV GeV GeV GeV GeV =10=10=10=10=10=10=10=10 66666666 (TP)(TP)(TP)(TP)(TP)(TP)(TP)(TP) QQQQQQQQ 4 too warmtoo warmtoo warmtoo warmtoo warmtoo warmtoo warmtoo warm 1010101010101010 Q mmmmmmmm 1010101010101010 m as violet lines how the yield changes according to ref. [ -7-7-7-7-7-7-7-7 1010101010101010 00000000 66666666 55555555 44444444 33333333 22222222 11111111 99999999 88888888 77777777 10

1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010 1010101010101010

R R R R R R R R (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) T T T T T T T T , our anomalous couplings coincide fully with the . The same as figure M > T ] for different values of the heavy quark masses. Note in any case that even without suppression, the DFSZ axin 58 inated by the decay term up to temperatures of the order of 10 insertion arise at lowestdiagrams order in at thermal two field loops, theory probably is needed a to full resolve this iss case, we show in figure ishing gluon mass. Since graphs with the one loop mass. In particular, we find no suppression at all for the Figure 10 KYC is in part supported byGovernment the (KRF-2008-341-C00008 Korea and Research Foundation No. Gra 2011-0011083).Planck Society K (MPG), the Korea MinistryGyeongsangbuk-Do of and Education, Pohang Scie City forGroup at the the support Asia of Pacific Center the for Inde Theoretical Physics (APC Acknowledgments The authors would like to thank theIts Galileo nature, Galilei origins Institut and prospects for detection” for hospit small heavy quark masses, results are in perfect agreement. ref. [ which our present studythe is loop, based, is practically not affected JHEP04(2012)106 h· · ·i (A.5) (A.2) (A.3) (A.4) (A.1) ). ··· , keeping in + in the early . i 2 ) is the axino ˜ a n ∗ i g ) is the number of → A.1 ∗ i = ··· g mework Programme ∗ + s g , ˜ a e of the Foundation for ]. j ed in table ment (MEST) (No. 2005- n → i 59 nsortium for Fundamental rted in part by the National , where i n 2 h.s. of eq. ( i -th particle’s number density ic density in detail. The time xinos are decoupled from the f. [ i e TH-Institute DMUH’11 (19- ) is the axino production from Γ( T in the thermal bath with cross h rel ) tial spins and summing over the v 2 P . j , i ) A.1 i X M n . dec is the i ··· i ) + ) and i s Y (90 j ( + n i / n n i ) ··· sHT ˜ a i . ¯ σ ∗ X ) n , g 2 + rel a i ˜ a 2 → v f + s 3 s ˜ a 3.2 n π 2 rel sHT j ( α v π ) → ≡ 4 + scat q i,j – 28 – i i , is described by the Boltzmann equation: Y ( Y ˜ a ··· σ Γ( n ) = h ) = h + i,j s X ( T ˜ a ( n dT dT = σ H R R → ˜ a T T Y j 0 0 Z Z + i = = is the entropy density, and normally ( σ 3 h dec ) and relative velocity scat i T i,j Y ∗ i,j Y s X ··· g = + 45) ˜ a ˜ a / 2 → π Hn particle which are in thermal bath with the decay rate Γ( j i + 3 + i = (2 ˜ a ( is the Hubble parameter, σ dt s dn H The relevant 2-body scattering cross sections are summariz Using the axino yield defined as Universe, the solution of the Boltzmann equation is effective massless degrees of freedom. The first term in the r. MRTN-CT-2006-035505. A Details of axinoIn relic this appendix, abundance we computation presentevolution the of computation the of the axino axino number rel density, 29 July 2011) during the completionResearch of Foundation this (NRF) work. grant funded JEK by0093841). is the suppo Korean LR Govern isPolish partially Science, supported by by thePhysics the Lancaster, under Welcome Manchester Programm STFC and grant Sheffield ST/J0000418/1, Co and by the EC 6th Fra the CERN Theory Division for hospitality in the context of th mind that the physical cross-sections are eq. ( where where production from the scattering process of particles the decay of final spins. Here wethermal neglect bath the and inverse their processes number since density the is a very small. denotes the thermal averaging including averaging over ini The explicit formulae for the thermal average are given in re section in the thermal bath. The second term in the r.h.s. of eq. ( Here JHEP04(2012)106 , 2 as nly T . . . t 2 (A.6) (A.8) (A.7) (A.9) g


 (A.11) (A.10) (A.12) 2 eff 2 eff 2 eff = /m /m /m 2 eff ) ) ) s s s m + + + 2 eff 2 eff 2 eff . , m 2 m m | ( ( (
 abc 2 eff f ence due to the massless | log log log ) -channel, it changes to /m


2 ) 3 3 u 3 t . s s s s for these processes. 2 process B with using massless | + n + + 2 - or + 4 σ + 4 abc t 2 eff 2 eff dt. f ts | 2 eff 2 eff 2 m | m )
( 2 n 2 m m 2 + 4 t s 2 2 s s M s 2 ) is obtained after integration over |
2 + s s , by log ) 2 s + 4 0 ( n s
− ts + 23
2 eff B s + 12 + 4 + 12 Z 2 4 eff M σ t s m t 2 2 eff + 7 + + 2 1 7 4 eff ]. Then we obtain finite and always positive 4 eff s sm 2 eff − 2 2 3 s 2 eff a 2 eff − – 29 – sm s 60 t 2 for the identical initial particles, m sm sm α ( / , m 6 + 6 4 πf 2 eff sm 2 2 4(2 − + 5 6 eff + 58 − + 11 sm + 9 + ) = m 4 eff 5 6 eff 4 eff s = 2 s 6 eff ( m 4 eff m 2 − m 2 n | 3 m (3 m ¯ − σ 3 24 B t 4 eff  − 3 2 +12 -channel. To regularize this IR divergence, we introduce, o 2 m M s | | s − u + 3 s + a − jk 4 −  = T s |  
2 - or |
s
t s 2 s B | ). Here we list the full cross-sections ¯ + + + M 2 | abc 2 2 eff | | A.6 f 2 eff 2 eff | a m jk abc m m 2 2) T f | -channel gluon propagators, an effective thermal gluon mass | / 2 u 2 2 s s (1 s ) is given from the matrix element, s or 16 16 12 ( n t σ 4. Process H: 3. Process G: 1. Process B: 2. Process F: We include the factor 1 Among the processes, B,F,Ggluon and exchange H in have the an infrared (IR) diverg Then ¯ given in the eq. ( cross sections. For example, thegluon squared propagator matrix is element for which is positive definite by definition. Then ¯ However introducing effective thermal gluon mass in the in the which is generated by plasma effects [ JHEP04(2012)106 ∼ eff 2 eff . We s/m (A.15) (A.16) (A.17) (A.14) eff , m ommons 2 T 5. So even . s/m 0 ≃ . We can see s s ∼ 2 g for the IR divergent 17 (A.13) . So with these formulas 0 11 > le where ) T al average larger than in the ( g to screening and physical de- s. On the other hand the full act substituting s lower than the real one due 2 mass. d reproduction in any medium,  g ntermediate gluon and therefore remain  , in the limit of large  , we use the expressions are given 3 6 eff  2 s 2 eff s 2 eff m 2 eff s s m m  m   O  O provide unphysical result for small O O + 2 + ) are positive for any value of + + 0 for 2 4 eff 2 2 | s 2 | | < m – 30 – A.12 A,C,D,E,I,J a abc 2 jk 4 7 abc 24 32 32 | f T f | because those reduce to: a | jk | 24 − 3 T | )] /s T 2 eff ( 2 ) to eq. ( m /g A.9 log[1 ∝ B ¯ σ This article is distributed under the terms of the Creative C . 2 , one finds e.g. for the B process In our numerical computations we use the full formulae above These formulas give back the expressions in table 2 Note that the B and H channels also contain diagrams without i 1. Process B: 3. Process G: 4. Process H: 2. Process F: T 11 2 mention here that those expressions in table that easily in the limit of large Attribution License which permits anyprovided use, the distribution original an author(s) and source are credited. the single cross-sections areprevious estimates. always positive and the therm processes, while for thein other table processes, Open Access. so they have the correctcoupling asymptotic of behaviour the correspondin intermediate gluon channels for large gluon if the total thermalto the cross-section negative may contribution resultexpressions of above positive, these from it eq. IR-divergent i ( channel and give negative contribution to the thermal averages. In f and this holds for all the interesting region up to the GUT sca g finite also in the limit of infinite gluon mass. JHEP04(2012)106 ] 19 , ]. , , , SPIRE ]. IN arXiv:0909.3293 ][ [ (2009) 105003 , Signatures of axinos , SPIRE 11 IN (2011) 039 ][ , 04 (2010) 036 (2010) 014 , k matter (2009) 193 06 ios 07 ]. ]. d F. 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