Jhep06(2010)036

JHEP06(2010)036 Springer June 8, 2010 April 6, 2010 May 25, 2010 : : : Received Published Accepted 10.1007/JHEP06(2010)036 doi: Published for SISSA by 1003.5847 Supersymmetry Phenomenology We reconsider thermal production of axinos in the early universe, adding: a) [email protected] E-mail: CERN, PH-TH, CH-1211, Geneva 23, Switzerland Dipartimento di Fisica dell’Universitàdi Pisa, Largo Pontecorvo 3, 56127, Pisa,INFN Italia – Sezione diLargo Pisa, Pontecorvo 3, 56127, Pisa, Italia Open Access times larger than previous computations. Keywords: ArXiv ePrint: Abstract: missed terms in the axino interaction;by b) thermal production masses; via gluon c) decays kinematically a allowed precise modeling of reheating. We find an axino abunance a few Alessandro Strumia Thermal production of axino dark matter JHEP06(2010)036 ] in the 1 of the 2 decays 10 . We here ∼ → RH 2 scatterings 3 T g gluino + axino ]. In supersym- → ], that manifests 2 1 → GeV [ 9 5 10 ; plus ii) thermally in the early ∼ > f NLSP ]. Depending on the model of super- /m 5 ]). It is thereby interesting to compute – 9 3 [ NLSP ]; [ a 8 Ω ˜ a – 1 – m = 2 ˜ a was just below the reheating temperature with a decay constant 5 T a ]; cold axino: [ 7 gets extended into an axion supermultiplet which also contains a 2 and the fermionic axino ˜ s 1 we show that axinos are also thermally produced by 1 we show that, beyond the well known axino/gluino/gluon interaction, we precisely model the reheating process. 6 ]; hot axino: [ 3 2 4 6 process gives the dominantstrong coupling contribution constant; in view of the large value superymmetry, that contributes atthat the produce axinos; same order to the usual 2 kinematically allowed by the gluon thermal mass. The gluon there is a new axino/gluino/squark/squark interaction, unavoidably demanded by in section in section in section c) a) b) As a result the axino production rate is significantly enhanced. universe when the temperature reconsider the thermal axinofollowing abundance, ways: improving on previous computations [ symmetry breaking, the axino canstable easily lightest be supersymmetric lighter particle than(warm (LSP) all axino: and other [ consequently sparticles, a becomingits Dark the cosmological Matter candidate abundance.lightest The sparticle (NLSP), axino such can that be Ω produced: i) from decays of the next to The strong CP problemat can low be energy solved as bymetric a the models Peccei-Quinn light the symmetry axion axion [ the scalar saxion 1 Introduction 3 Thermal axino production rate 4 Thermal axino abundance 5 Conclusions Contents 1 Introduction 2 Axino couplings JHEP06(2010)036 , ]. π . 1 4 q 10 / and 2 (3.1) (2.2) 3 g . The 2 [ . ]). We ··· 1 = ˜ q . + ) + a ∼ > 12 3 a T/m , ˜ a 3 T ˜ g α θ g ∗ 11 / 2 ˜ q D i √ a ]; [ ˜ q ¯ ˜ g . 8 X 3 2 + 2 3 . g + 4 g √ 1 1) gives the following − a / in the super-multiplet ) GeV, lies in the region ln D a = , leaving a ln ]). It contributes to the 2 ia a 3 g 10 3 a g T g D + 12 3 4 10 2 – . g s ]: 2 scatterings listed in table ] at leading order in the strong ∼ − 10 i.e. 10 ∼ = ( → ]; cold axino: [ 10 + h.c. (2.1) T T a µν ]; [ ,D 7 m A a 8 ) = 32 G 3 a g 7 becoming negative for W and gluino ˜ a µν . ( D a 0 m a 85 at -channel gives an infinite cross-section a G . t ( ˜ g G 0 ∼ > s ¯ ˜ a HTL 3 2 ≈ g θ AW 2 − 3 )) + d g a a ˜ g -term, where the sum runs over all squarks ˜ ˜ g ]; hot axino: [ – 2 – Z 5 ,F 5 is: 6 ) D γ ]; cold axino: [ γ 3 3 ] is the axion decay constant, related in a well known ¯ ˜ g 7 µ g α ν µ γ ( f πf 2 a 8 D , γ 2 ¯ ˜ g √ µ µ ( HTL γ µ − ]) approximation ( [ F D 2 = a µν iθθσ 14 f , 4 3 + 7 eff − g ¯ ˜ aG π 13 6 ] a µν i L ), the axion supermultiplet can have extra non-minimal couplings T ˜ ]; hot axino: [ a µν G + 256 6 G 2.1 a µν = µν G ( 2 iσ a → . This roughly amounts to compute the 2 2 − 3 γ 3 a g πf α 8 D [ θ = + illustrates that the physical value, is the super-space coordinate, eff a g 1 θ L In addition to eq. ( By expanding in components and converting to Dirac 4-component notation we get: = ˜ a This production rate unphysically decreases for Figure with thermal effectsgluons: ignored indeed everywhere a but masslessbecause gluon in it exchanged the mediates in propagator the adivergence of is long-range cut Coulomb-like virtual off force; intermediate Hard by Thermal the the Loop resulting thermal (HTL infra-red [ massresult of logarithmic for the the gluon, space-time density of scatterings into axinos [ 3 Thermal axino productionThe rate axino production thermal rate hasgauge been coupling computed in [ inserting the explicit value of the strong to the electroweak vectors(warmonly axino: consider [ theaxino presumably production dominant rate. strong interaction contribution to the thermal The first term is the usualsaxion axion coupling coupling to and the gluon; bothterms the second are in term accompanied the is the by lower corresponding rowliterature(warm couplings are axino: to the [ gluinos axino andaxino couplings; to production the squarks. latter rate term at The was the not same considered order in as the the well known third term, as can be seen by where way to the parametersencodes of the each RGE axion renormalization model of at the full hand. operator. The RGE evolution of The effective coupling between thethe axion supermultiplet strong gauge vectors,W described by the gluon 2 Axino couplings JHEP06(2010)036 also 1 ], that we 14 2 scatterings 2 process has summed over all → ↔ 2 f 4 π . Subsequent higher . Indeed this process γ 3 0 128 0 g (continuous lines); they / tC 6 sC 3 0 0 0 0 tC sC g 2 F 2 subtracted − | − factors. Our goal is including A , because a 1 | 2 π /π 3 g ) ) ) ) 2 /t /stu /s /t /s ]. ) 0 2 ) 0 2 2 2 ] for gravitino thermal production) the 2 ) s t t s at next order in 2 scatterings. Thereby such decay gives /t 2 14 /s 15 2 u 2 γ ) (dashed line) is unreliable. Figure 0 s 0 0 t + 2 + 2 3 + 2 + 2 ↔ + g t 2 full s t + s + | ( tC tC sC 2 t 2 2 s t 2 – 3 – gluino + axino (3.2) ( ( A 0 + 2 0 + 2 + 2 − − + 2 | + HTL t s C s t C = 48 after summing over all quarks. → 2 ( ( F ( ( 4 4 0 0 ], the axino production rate is precisely defined in s 2 to the axino production rate | denote gluons, gluinos, quarks, squarks. The gauge factors ( C C − − C C a 15 ij ˜ 4 4 2 q 2 2 C ) 1. T is replaced by the function 8 | − 3 − gluon q − ∼ ˜ g, q, πg 3 P HTL g, g ˜ ˜ ˜ ˜ ˜ ˜ a a a a a a ˜ ˜ ˜ ˜ a a a a = F ˜ ˜ ˜ ˜ ˜ ˜ g g g q g q q g q g 0 C → → → → → → → → → → ¯ ˜ ¯ ˜ ˜ ˜ ˜ q q g g g g process ˜ ¯ ˜ ˜ ˜ q qq q ˜ q qg gq q g gg g 2 scatterings. Thereby, in order to avoid overcounting, 2 I J F E B C A H D G = 24 and 2 → 1 and differ at | ), taking into account that the gluon and gluino thermal masses break Lorentz abc f | 3 3.2 g . Squared matrix elements for axino production in units of = C Thermal field theory is just a tool to describe the collective effect of scatterings, and Proceeding along the lines of [ To improve the computation going beyond the leading-order HTL approximation, we , the decay rate is enhanced by a phase space factor 3 invariance, and that actually athe continuum gluon of and ‘masses’ gluino is thermal present, spectral as densities precisely [ describedthe by decay diagram indeedlowest-order resums 2 an infinite subset of scatterings, and in particular some terms of the imaginarycompute using part the of resummed finite-temperature theloop. propagators thermal for axino The gluons and propagator resultingprocess gluinos at expression in ( the one can loop be [ interpreted as the thermal average of the decay g a phase space largera by correction this of amount relativeorder than order corrections 2 ( should bethe suppressed enhanced by higher the orderfeasible usual terms, because a and decay this is finite-temperature a simple computation enough is process. practically new decay process first contributes to theis axino made production kinematically rate allowed by the gluon thermal mass. Despite being higher order in where the leading-order rateillustrates function our final result: agree at notice that (analogously to what discussed in [ Table 1 polarizations and gauge indices. equal JHEP06(2010)036 ) ], 10 15 3.3 (3.3) 10 9 10 is a model- ˜ q was initially 8 ˜ a a 10 m µ T ∗ production rate, S ˜ q : GeV 7 µ µ χ in ∂ 10 ˜ q S gives explicit values X a 1 6 . By comparing with ˜ g F HTL 3 10 3 F g M 1. Temperature 5 − ∼ 10 a ]. Table 3 2 diagrams. What remains are ˜ g g 5 , the axino mass ], which asymptotically agrees with 4 15 γ GeV. The lower dashed curve shows → ] 10 Pl 10 3 ν ¯ − M 6 , γ 3 3 − µ 10 √

γ 10 5 0

[

30 25 20 15 10

F function Rate 10 F/ a µν ∼ G = 2 diagrams [ i 4 T 2 / → – 4 – 3 − 1.4 m 3 factors appear because all divergent Coloumb-like M

3 1.2 T = 10 GeV /t = ]. While the goldstino mass is predicted in terms of µ 18 6 term as S ], as well as giving a non-vanishing subtracted contribution. T = 10 GeV 1.0 , and behaves unphysically for µ 3 3 F at temperatures ∂ 10 g

10 g

10 3 that describes our result for the axino production rate. The arrows g 0.8 F coupling F HTL 0.6 F Strong µ S F 0.4 µ 2 in Hard Thermal Loop approximation from [ √ ¯ χ ∂ 0.2 HTL = F . Rate function ) we notice that the Goldstino and the axino couple to the same combination int 0.0 2.1 5 0

20 15 10 25 30

Actually there is a stronger connection between the axino and the gravitino thermal F function Rate eq. ( of operators.missed Even and in finally thethe noticed supersymmetry-breaking Goldstino by case, [ dependent the free parameter. second operator in induced by its coupling to the divergence of the MSSM super-current having including only the contribution from the gluino mass G, I vanish:is a not single the diagram casecontributes, contributes, for changing such scatterings the H that coefficientthe and no of corresponding J, table the interference 1 where first terms of the [ term exist. second among axino parenthesis This interaction withproduction in respect rate. eq. to The latter ( contains the contribution of the Goldstino which includes the modulus squaredinterferences of between single different Feynman 2 Feynman 2 for the total andrate subtracted is axino scattering infra-red rates.scatterings convergent: are Unlike no the included total 1 in rate, the the decay subtracted diagram. Subtracted rates for processes C, D, E, the result our result in the limit of small must be added subtracting the contributions already included in resummed decay [ Figure 1 indicate the MSSM values of JHEP06(2010)036 2 R 1. ρ → (3.4) (4.1) ∼ 3 g is approx- RH T 2 3 g ] for the Goldstino and of the reheating 2 6 π ˜ a 15 · m 29 gives the correction . 29 ], our description of the . 1 such that thermally produced 19 , + 1 f GeV 3 f 12 16 , 10 1 = the result of [ - = 15 10 F , f πf γ 10 4 GeV / ([ = 3 in ], our description of the reheating process ) 2 ˜ a ˜ a α 3 - n 2 16 g 10 ( mass √ Hn F GeV ]; the second term is the contribution of 2 2 – 5 – ↔ 11 axino f + 3 6 GeV 15 (computed including only the energy density 7 10 , behaves physically for physical values of 3 - ˜ a T , as function of the axino mass 13 π 4 /F 3 1 = , and its numerical factor 1 10 dt Pl 10 3 g f dn DM 1 256 = M /M f 3 = = Ω / 4 2 - ˜ a R 10 → 5 4 9 8 7 6 sub 2 ]) πρ

10 10 10 10 10 10

8 temperature reheating GeV in 17 + p 2 → ≡ 1 , plotted in figure γ R F H = . γ RH T was numerically computed in [ . Contour-plot of the value of the axion decay constant 3 f Our result for In conclusion, after converting imately defined as the maximalmore temperature precisely of the as universe the (‘instantaneousthe temperature reheating’), Hubble at or which rate theof inflaton the decay thermal rate bath), becomesluted such smaller by that than energy whatever injection happened from earlier inflaton decay at ([ higher temperatures got di- The axino abundance can now bemann computed equation by integrating for the the relevantreheating cosmological axino process Boltz- number follows [ density as well as the equations that describe reheating. The reheating temperature the Boltzmann approximation. 4 Thermal axino abundance where subtracted scattering rates ofcoming table from the appropriate Fermi-Dirac and Bose-Einstein distributions with respect to thermal production rate is translated into the axino production rate: Figure 2 axinos are all cold darktemperature matter, Ω JHEP06(2010)036 ]. DM 15 (4.2) (4.3) Ω ˜ a 3 3 g . ] for the axino , by a factor of GeV in order to ln 15 1 2 3 ) we assumed that 12 g 10 20 4.3 ∼ < ≈ f ) 3 g term was missed in previous ( (the analytic approximation ) and ( ˜ q ] based on the Hard Thermal 1 ∗ ˜ q 4.2 10 ˜ g a : ,F s 2 determines the axino rate as described by ]). In eq. ( GeV f 745 realistic reheating . F 25 such that thermally produced axinos have an 11 0 1 instantaneous reheating f 10 ) plotted in figure – 6 – ( 3 ], and the resulting phenomenology was studied g × ( 20 F coupling, the other ˜ RH GeV T RH 1, which gives unphysical results at the physical value 4 = T ˜ gG 006 [ and T . a

10 0 s RH 3 ˜ a R ± γ g T 1). The axino energy density must be equal or smaller to the H m GeV ) 110 ≈ = . 3 ) GeV. The function 3 g 7 ( g RH , otherwise the axino reaches thermal equilibrium, giving Ω we plot the values of = 0 F T s 4 (10 ). 3 2 3 2 g − This article is distributed under the terms of the Creative Commons 4 T h

4.3 24 . ˜ a s DM n 8 10 . = 10 1 = 1 we derived the axino coupling to the strong sector, finding that it is described by ]). The two different definitions give the following solutions [ 2 10 eV. RH ≈ 2 h ) and ( ]. In figure T 17 1. As a consequence we find an enhancement, plotted in figure ˜ a renormalized around ˜ eq a 24 Ω 3.4 3 n ∼ ˜ – a g 3 m 21 g ˜ a We thank L. Covi, V.S. Rychkov, C. Scrucca andOpen G. Villadoro. Access. Attribution Noncommercial License whichand permits reproduction any in any noncommercial medium, use, provided distribution, the original author(s) and source are credited. 6 (3) at eq.s ( Acknowledgments This makes the axino interaction fullycould analogous infer to the the Goldstino axino interaction,Such thermal such result production that goes we rate beyond fromLoop the the (HTL) leading Goldstino order approximation rate, computations computedof [ in [ 5 Conclusions In section two terms: one is the wellstudies, known although ˜ they contribute at the same order to the thermal axino production rate. avoid a too large axion DMof abundance, the unless axion the potential initial (forn axion a vev recent is study close see to [ thefor minimum with is appropriate around DM density Ω in [ abundance equal to the observed DM abundance. We recall that sity is number abundance normalized to the entropy density Using the numerical factor appropriate for realistic reheating, the present axino mass den- follows [ JHEP06(2010)036 ] , B (1999) 08 (1984) 82 ] (2007) Phys. Phys. Rev. B 198 , Nucl. Phys. , (1983) 135 , JCAP , Phys. Rev. Lett. D 75 37 (1999) 83 B 138 Phys. Lett. , hep-ph/0310123 , (2009) 105003 [ (2007) 377 11 Nucl. Phys. B 447 , 170 Phys. Rev. Lett. hep-ph/0002154 Phys. Rev. , [ Phys. Lett. , Towards a complete theory of (2004) 89 , ]. Nuovo Cim. Lett. Axino dark matter and the Phys. Lett. B 685 , , New J. Phys. (2000) 41 , 87 SPIRES ][ , Addison-Wesley, U.S.A. (1990). Astrophys. J. Suppl. Unstable photino mass bound from cosmology , Cosmological implications of axinos Nucl. Phys. Wilkinson Microwave Anisotropy Probe (WMAP) , – 7 – ]. ]. Axinos as cold dark matter ]. Axino dark matter from thermal production , Cambridge University Press, Cambridge U.K. (2000). ]. ]. hep-ph/0402240 Is it easy to save the gravitino? Thermal production of gravitinos SPIRES Simple effective lagrangian for hard thermal loops SPIRES [ CP conservation in the presence of instantons [ The early universe Axions and supersymmetry Is axino dark matter possible in supergravity? ]. ][ Broken global symmetries in supersymmetric theories ]. ]. ]. SPIRES Nucl. Phys. Proc. Suppl. , ][ Supersymmetry and the strong CP problem ]. SPIRES SPIRES ]. [ Axinos as dark matter particles Interactions of a single Goldstino ][ SPIRES ]. (2004) 003 SPIRES SPIRES SPIRES (1984) 346 ][ [ [ ][ 06 SPIRES SPIRES [ [ Thermal field theory ]. (1982) 451 SPIRES hep-ph/0405158 B 139 Axion searches [ [ JHEP collaboration, D.N. Spergel et al., ]. ]. ]. , hep-ph/0701104 (1991) 447 [ (1992) 1827 hep-ph/9905212 B 112 SPIRES (1992) 103 [ [ (1977) 1440 hep-ph/9805512 arXiv:0902.0769 SPIRES SPIRES SPIRES [ 265 WMAP three year results: Implications for cosmology thermal leptogenesis in the[ SM and MSSM D 45 075011 (2004) 008 CMSSM [ 4180 276 [ B 358 Phys. Lett. Lett. (1982) 102 38 [ L. Covi, J.E. Kim and L. Roszkowski, T. Goto and M. Yamaguchi, K. Rajagopal, M.S. Turner and F. Wilczek, J.E. Kim, A. Masiero and D.V. Nanopoulos, K. Tamvakis and D. Wyler, H.P. Nilles and S. Raby, J.M. Frere and J.M. Gerard, R.D. Peccei and H.R. Quinn, P. Sikivie, M.Y. Khlopov and A.D. Linde, E.W. Kolb and M.S. Turner, G.F. Giudice, A. Notari, M. Raidal, A. Riotto and A. Strumia, T. Lee and G.-H. Wu, E. Braaten and R.D. Pisarski, M. Le Bellac, V.S. Rychkov and A. Strumia, L. Covi, L. Roszkowski, R. Ruiz de Austri and M.L. Small, Covi and J.E. Kim, A. Brandenburg and F.D. Steffen, [8] [9] [6] [7] [3] [4] [5] [1] [2] [19] [20] [16] [17] [18] [13] [14] [15] [11] [12] [10] References JHEP06(2010)036 ] ] ]. SPIRES [ ]. hep-ph/0008139 [ arXiv:0710.3349 [ SPIRES [ ]. arXiv:0910.1066 , SPIRES (2008) 016 ][ (2000) 125001 04 Determining reheating temperature at D 62 arXiv:0910.0333 JHEP , Cosmological implications of supersymmetric , – 8 – Cosmological constraints on a Peccei-Quinn flatino as Phys. Rev. arXiv:0711.3083 , [ Axion cosmology revisited ]. (2008) 009 Fine-tuning favors mixed axion/axino cold dark matter over SPIRES 03 ][ JCAP , ]. ]. astro-ph/0603449 SPIRES SPIRES axion models neutralinos in the minimal supergravity model the lightest supersymmetric particle [ colliders with axino or[ gravitino dark matter [ O. Wantz and E.P.S. Shellard, M. Kawasaki, K. Nakayama and M. Senami, H. Baer and A.D. Box, E.J. Chun, H.B. Kim and D.H. Lyth, K.-Y. Choi, L. Roszkowski and R. Ruiz de Austri, [25] [23] [24] [21] [22]