arXiv:1104.0692v1 [hep-ph] 4 Apr 2011 hs otiuin a epeeti h omof form the in present be may contributions these where nti basis this In h xo ietyt h uesmer raigfield, breaking supersymmetry couple the that to operators directly higher-dimensional axion one the of sec- grounds, way PQ the by general into importance tor On seep utmost to of breaking value. supersymmetry is typical expects it its mass, axino ascertain the to of axion. size [2] the DFSZ on the of can su- case there other PQ the axion, and in a coupling [1] axino as the KSVZ axino be perpartners, between the only to couplings of the additional defined case be be the is may in axino (1) as present, the Eq. While basis this sym- singlet. (PQ) In Peccei-Quinn realized metry. nonlinearly the under hr h uefil otiigtesxo,ain and axion, saxion, as the defined is containing axino superfield the where ihasrnt nesl rprinlt h xo de- axion the to proportional constant, inversely cay strength a with axion. QCD we explore supersymmetric paper, and the this of proposals cosmology In theoretical the LHC. these the of for both embrace implications and of theoretically problem wealth hierarchy a a gauge Like- the offers to resolution probes. supersymmetry motivated laboratory weak-scale to and amenable astrophysical wise, uniquely of is variety which problem a CP strong the to pnigculn fteQDaiot h lio Ex- form the gluino. takes the interaction to this axino superspace, in QCD pressed the of coupling sponding ic h omlg fteetere eed crucially depends theories these of cosmology the Since ycntuto,teQDainculst h gluon the to couples axion QCD the construction, By solution elegant extraordinarily an is axion QCD The F 3 nttt o h hsc n ahmtc fteUies,U Universe, the of Mathematics and Physics the for Institute stesprymtybekn cl.While scale. breaking supersymmetry the is L A = reult hto h rvtn nteasneo naua fi lim pr stringent unnatural quite axino of a absence QCD provide by the given and together inflation in gravitino and gravitino space, thermal the the parameter from that of bounds implies that consequence, analysis to general equal analogous A or fully production. is thermal that via problem axino cosmological serious yti eut eepoetecsooyo rvtn S and LSP gravitino of cosmology the explore we result, this by he elsi cnro o akmatter. dark for scenarios realistic three A 2 ( = f 1 hoeia hsc ru,Lwec eklyNtoa Lab National Berkeley Lawrence Group, Physics Theoretical ervsttecsooyo h uesmercQDain hi axion, QCD supersymmetric the of cosmology the revisit We eklyCne o hoeia hsc,Uiest fCal of University Physics, Theoretical for Center Berkeley hfsb nabtayiaiayconstant imaginary arbitrary an by shifts a √ 8 .INTRODUCTION I. uesmer hnrqie corre- a requires then Supersymmetry . X πf 2 s α a + 3 = Z ia x ) d / + 2 √ θAW √ lffr Cheung, Clifford + 2 T 2 R θη a √ < W + 2 10 θ a θ h omlgclAioProblem Axino Cosmological The a ˜ 3 2 h.c. + − + F, 10 θ 2 6 F , e o naindcycntn of constant decay axion an for GeV A ,2 1, . il Elor, Gilly (3) (2) (1) ,2 1, o nteeetv edter below theory field opera- effective higher-dimensional the a in of tor existence the by derstood tr rnfrsprymtybekn ffcso order of oper- effects these breaking cal- m Since supersymmetry the transfer [3]. from supergravity ators irreducibly ultra- of arise dynamics unspecified also culable by they physics, induced violet operators slop” “Planck ieryraie Qsmer,teaiohsams of mass a non- has the turn, axino by In the massless order symmetry, be order. PQ to realized this ensured linearly is of axion difference the mass because a acquire axino ashsalwrbudo order of bound axino lower the a Hence, has symmetry, mass dynamics. PQ Planck-scale full from the expected preserves exactly which rvt ffcswihtpclyyedtesm eutas result same the yield typically which super- theories, effects through these mediated gravity in still is even breaking Nevertheless, supersymmetry the to [12–15]. from used sector breaking are PQ supersymmetric dimensions sequester extra which effectively in scenarios certain generated is and physics. symmetries Planck-scale all uncontrolled by by allowed gravitin is the which central order mass of a fine-tuning, contribution embodies irreducible barring an (5) acquires paper: Eq. this NLSP. of axino claim and grav- with LSP considering breaking itino supersymmetry for low-scale motivation of theoretical theories a provides which 2 1 rtdb lnksaednmc 57.Se[–1 o possib for [8–11] gen- See are they difficulty. [5–7]. if this dynamics even to problem, scale resolutions Planck destabilize CP strong by easily the erated very to can solution operators axion breaking PQ Explicit eetyasmlreetv hoyhsbe osdrdi [4 in considered been has theory effective similar a Recently 3 oceey hssml hsclagmn a eun- be can argument physical simple this Concretely, n arneJ Hall J. Lawrence and oal xeto oti ipeagmn rssin arises argument simple this to exception notable A / 2 Z m ∼ iest fTko ahw 7-58 Japan 277-8568, Kashiwa Tokyo, of niversity otegaiiopolmo overclosure of problem gravitino the to d 3 to h eetn eprtr after temperature reheating the on it 4 F/m / C xn a asgetrthan greater mass a has axino QCD fri,Bree,C 42,USA 94720, CA Berkeley, ifornia, θ 2 rtr,Bree,C 42,USA 94720, CA Berkeley, oratory, xn LPa low at NLSP axino u oPac-cl corrections. Planck-scale to due ( A etnn rsqetrn.A a As sequestering. or ne-tuning Pl dcinaecmlmnayin complementary are oduction f + a hihigteeitneo a of existence the ghlighting noteP etr h xo n the and axion the sector, PQ the into A 10 = † m ) 2 m 9 Pl ( X − a ˜ ,2 3 2, 1, 10 + & 12 X e.Motivated GeV. m T † ) R 3 / n present and 2 ∼ , 1 2 f m a 1 3 / , h xn mass axino the 2 a ˜ a ˜ + 2 , . . . n is and ]. (4) (5) the le o 2 in Eq. (4) [16]. In this paper we will assume that these Next, let us consider the effect of higher-dimensional more complicated dynamics are not at play. operators of the form As we will see, the bound in Eq. (5) has an enormous † † effect on early universe cosmology. In particular, it im- 4 λiΦi Φi(X + X ) λifi † † d θ = (F F +F Fi)+..., (7) plies the existence of a cosmological axino problem which Z Λ Λ i is similar to the well-known gravitino problem [17] but 3 which occurs in a complementary region of m3/2. To- which couple supersymmetry breaking to the PQ sector. gether, the combined bounds from axino and gravitino Here λi is an order unity dimensionless coupling and X production completely exclude the possibility of a high is as defined in Eq. (3). The parameter Λ is the mass TR, as shown in Fig. 1. In turn, this rather unequivocally scale of the heavy particles which couple the PQ and nullifies the viability of high-scale leptogenesis [18] while supersymmetry breaking sectors and, as discussed in the still permitting low-scale models of soft leptogenesis [19] introduction, it is at most the Planck scale, Λ . mPl. and testable theories of asymmetric freeze-in [20, 21]. It is clear that the only fermion mass terms gener- Let us now briefly outline the remainder of this pa- ated by this operator are a mixing between ψi and η, per. In Section 2, we present the theoretical rationale for the fermionic component of X. Morally, this can be un- Eq. (5), both in general and for a canonical supersymmet- derstood as a rather surprising mixing term between the ric axion model. We then go on to explore the thermal axino and the goldstino. Such a mixing is strange, but production of axinos in the early universe in Section 3. by including a dynamical supersymmetry breaking sector Here we take note of a novel regime in which the domi- for X and diagonalizing the fermion mass matrix in the nant mode of axino production arises from the decays of presence of Eq. (7), one goes to mass eigenstate basis and superpartners still in thermal equilibrium, i.e. freeze-in expectedly finds a massless goldstino and a heavy axino [22]. Freeze-in of gravitinos and more generally of hid- with a mass of order den sector dark matter was considered in [23] and [24, 25]. F By combining the bounds on overclosure from gravitino ma˜ , (8) and axino production, we can then precisely quantify the ∼ Λ seriousness of the cosmological axino problem. We go on as shown explicitly for the very simple theory described in Section 4 to consider the mixed cosmology of graviti- in Appendix A. nos, axinos, and saxions. Applying bounds from overclo- One can reach the very same conclusion through a less sure, structure formation, and big bang nucleosynthesis technical, more physical argument. In particular, the (BBN), we determine the values of m , m and f for 3/2 a˜ a right-hand side of Eq. (7) manifestly induces a non-zero which the cosmological history is viable. Finally, we con- value for the auxiliary field F , clude in Section 5. i λ f F F = i i + ..., (9) i − Λ II. SUPERSYMMETRIC AXION THEORIES which mediates supersymmetry breaking effects propor- To begin, let us provide a more rigorous justification tional to Fi directly into the scalar potential of the PQ for the naturalness argument that implies Eq. (5). Ev- sector. Inserting appropriate powers of the characteristic ery supersymmetric axion theory can be described by an scale of the PQ sector, fi, we find that the mass scales ensemble of interacting fields and vacuum expectation values in the PQ sector shift by an amount which is of order F/Λ. Hence, one should 2 Φi = φi + √2θψi + θ Fi, (6) expect a mass splitting between the axion and the axino of order F/Λ which implies Eq. (5). whose dynamics induce vacuum expectation values, This argument can be restated in another way. In par- Φ = f . In the supersymmetric limit, the saxion, ax- ticular, apply a field redefinition h ii i ion and axino are linear combinations of Re φi, Im φi and ψ , respectively, and are massless as a consequence λiX i Φ Φ 1+ , (10) of the nonlinearly realized PQ symmetry. The entirety i → i Λ of our discussion will occur in the language of linear fields, Φi, so let us briefly note that the field A discussed which removes the connector interaction in Eq. (7) at in the introduction is a linear combination of the fields leading order in 1/Λ at the expense of introducing X 2 Ai/fi 4 Ai = si + iai + √2θa˜i + θ FAi where Φi = fie . directly into the PQ sector dynamics. The effect of X will be to explicitly generate an axino mass.
3 As in the case of the gravitino problem, this axino problem can be evaded if R-parity is broken. interactions involving only a single Φi. However, this quadratic 4 2 − 2 Naively, the θ component of Φi contains a term quadratic in the term appears in the combination (FAi a˜i /2), and so can always axino which can produce what appears to be an axino mass in be removed by a shift of the auxiliary field. 3
For example, consider the canonical supersymmetric 1010 1 axion theory defined by canonical Kahler potential and a superpotential 109
2 W = κ Φ (Φ Φ f ), (11) 108 3 1 2 − where κ is a dimensionless coupling, and a straightfor- 107 ward calculation shows that the vacuum is stabilized at 2 5 6 f1f2 = f and f3 = 0 in the supersymmetric limit. D 10
Consider the effect of the higher-dimensional operators GeV @ R in Eq. (7) for the symmetrical case λ1,2 = λ and λ3 = 0. T 105 After the field redefinition in Eq. (10), a simple calcu- lation shows that the vacuum is slightly shifted so that 104 f = λF/κΛ. This induces a mass for the axino equal 3 − to λF/2Λ from the superpotential, which again accords 103 with Eq. (5).
2 Before we continue on to cosmology, let us comment 10 1 briefly on some of the existing literature on the mass of the axino. While statements are frequently made to the 101 -6 -5 -4 -3 -2 -1 0 1 2 3 effect that the axino mass is highly model dependent, 10 10 10 10 10 10 10 10 10 10 m @GeVD we disagree. The authors of [16] and [28] computed the 32 axino mass in a variety of settings and verified Eq. (5) and Eq. (8) in all cases. The only exceptions resulted 2 FIG. 1: The red contour is Ω3/2h = 0.11 from gravitino from the imposition of special relations among disparate 2 production alone, and the blue contour is Ωa˜h = 0.11 for 12 and unrelated parameters to yield a lighter axino. We axino production alone with ma˜ = m3/2 and fa = 10 GeV. conclude that in generic theories the axino mass is given For larger values of ma˜ and smaller values of fa, the blue by Eq. (8) and the only model dependence is the value contour is translated to the left. The green contour can be 2 of Λ, which has an upper bound of mPl. interpreted in two ways: (i) as the (Ω3/2 + Ωa˜)h = 0.11 contour for combined gravitino + axino co-dark matter when both are cosmologically stable for ma˜ = m3/2; (ii) as the 2 III. AXINO COSMOLOGY Ω3/2h = 0.11 contour for gravitino dark matter when the axino is unstable and decays sufficiently quickly to gravitinos, for any value of ma˜ > m3/2. Observe that a high reheating Given a proper theoretical justification of Eq. (5), let 5 temperature, T > 3 × 10 GeV, is unambiguously excluded us now consider the early universe cosmology of the ax- R in both cases. The superpartner spectrum is taken to be ino. Like the gravitino, the axino is produced through {m˜b,mw˜ ,mg˜} = {100 GeV, 210 GeV, 638 GeV} and univer- thermal scattering and decay processes. As observed by sal scalar masses at the GUT scale equal to 500 GeV. Strumia [29], one can trivially compute axino production by translating every equation relevant to gravitino pro- duction by the simple replacement Moreover, as in the case of gravitino cosmology, one dis- m √2α covers regions where axino production arises dominantly g˜ 3 , (12) from freeze-in and scattering, corresponding to the ver- F ↔ 4πfa tical and sloped portions of the contours, respectively. while shutting off all scattering and decay processes in- Note that the axino contours in Fig. 2 are “flipped” from volving gauginos and scalars not accounted for in Eq. (1). the usual contours corresponding to gravitino overclo- In Fig. 2 and Fig. 3, we have numerically plotted sure, simply because the scattering and decay rates for 2 contours of Ωa˜h = 0.11 for mg˜ = 638 GeV and dif- axinos are independent of the axino mass and fixed by ferent values for the axion decay constant, assuming fa. the axino to be cosmologically stable. As one can see, for axion decay constants within the “axion window”, 9 12 10 GeV < fa < 10 GeV, the bounds on TR are quite A. Cosmological Axino Problem stringent. For a GUT scale axion decay constant corre- sponding to the “anthropic window”, the bound is weak. The cosmology of gravitinos and axinos has been stud- ied in great detail by numerous authors over many years [31]. Nevertheless, to our knowledge the existence of a se- rious cosmological axino problem has never been clearly 5 Our results hold irrespective of the precise dynamics which break supersymmetry—indeed, one obtains the correct answer even discussed. In particular, the red and blue curves in Fig. 1 when treating X as a non-dynamical spurion of supersymmetry are bounds on TR from gravitino and axino production breaking. overlayed, assuming that they are co-dark matter candi- 4
1010 1 1010 1
109 109
108 108 1
107 107 D D 6 6 10 10 1 GeV GeV @ @ R 5 R 5 T
10 T 10
104 104
103 103 1
102 102
101 101 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 ma@GeVD ma @GeVD
2 2 FIG. 2: Contours of Ωa˜ h = 0.11 for the KSVZ [1] model, FIG. 3: Contours of Ωa˜ h = 0.11 for the DFSZ [2] model, where axinos couple minimally, for mg˜ = 638 GeV. The where axinos have additional couplings to squarks, for mg˜ = {red, orange, yellow, green, blue} contours correspond to 638 GeV and mq˜ = 500 GeV. The new decayq ˜ → a˜ + q, log10 fa/GeV = {9, 10, 11, 12, 15}. results in a larger freeze-in region, where the contours are nearly vertical, as compared to Fig. 2. The contour colors correspond to the same values of fa as in Fig. 2. dates and ma˜ = m3/2. This is a conservative assumption to make because as shown in Eq. (4), the axino acquires ˜ an irreducible contribution to its mass of order the grav- where G is the physical goldstino. Note that this operator itino mass which cannot be eliminated without arbitrary respects the PQ symmetry because the axion is deriva- fine-tuning. On the other hand, if for whatever reason the tively coupled. The above operator mediates the decay axino is heavier than the gravitino mass, then this only of the axino to the gravitino and axion with a lifetime harshens axino bounds depicted in Fig. 1, since the axino m 2 5 contour is translated to the left by a factor m /m . 9 3/2 GeV a˜ 3/2 τa˜ 10 sec . (14) Fig. 1 illustrates a primary claim of this paper: the ≃ × MeV ma˜ combined gravitino and axino problems entirely exclude the possibility of a high reheating temperature. The upper This lifetime ranges over many orders of magnitude in 5 12 the m3/2,ma˜ plane, yielding a broad spectrum of cos- bound on TR is about 3 10 GeV for fa = 10 GeV, and × mological histories which we now explore. is even lower for lower fa. Thus, supersymmetry plus the axion solution to the strong CP problem are strongly at odds with theories of high TR and high-scale leptogenesis, 9 A. Three Scenarios for Dark Matter which typically require TR > 10 GeV. The bound on TR is greatly relaxed for very large fa, as could occur with anthropic selection of a small axion misalignment angle. The cosmological history varies substantially in the ma˜,m3/2 plane, as depicted in Fig. 4 and Fig. 5 for 9 12 fa = 10 GeV and 10 GeV respectively. The grey re- gion corresponds to m < m , which is disfavored by IV. GRAVITINO AND AXINO COSMOLOGY a˜ 3/2 the naturalness argument leading to Eq. (4), while the blue regions labeled by Ia, Ib, II, and III are consistent Finally, let us consider cosmology in theories with grav- with existing experiments. 18 itino LSP and axino NLSP, keeping careful track of the Region I of these figures, τa˜ & 10 sec, corresponds decays of axinos to gravitinos and cosmological limits on to a “Cosmologically-Stable Axino”. In this regime, both axinos and saxions. The operator in Eq. (4) gener- gravitinos and axinos are co-dark matter with ma˜Ya˜ + ates an axino-axion-gravitino coupling of the form m3/2Y3/2 = ρ0/s0, where m a˜ ˜ µ † ρ0 −10 Gσ a˜ ∂µa, (13) = 4 10 GeV (15) m3/2mPl s0 × 5
104 104 III III 103 103
102 102 II
101 101 II
100 100
- - D 10 1 D 10 1
GeV GeV Ib @ @ a -2 a -2 m 10 m 10
10-3 Ib 10-3 Ia
10-4 10-4 Ia 10-5 10-5
10-6 10-6
10-7 10-7 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104
m32@GeVD m32@GeVD
12 FIG. 4: Light blue regions in the (m3/2,ma˜) plane have realis- FIG. 5: Same as Fig. 4 but with fa = 10 GeV. Region tic dark matter cosmologies with a gravitino LSP and an axino II, corresponding to sub-dominant hot gravitino dark matter 9 NLSP for fa = 10 GeV. Regions Ia and Ib correspond dark from axino decay, has increased. The superpartner spectrum matter comprised of both stable gravitinos and axinos, with is taken to be identical to that of Fig. 1. gravitinos or axinos dominating the dark matter abundance, respectively. Region II is bounded by warm dark matter con- straints and limits on relativistic species. In Region III the are warm and will over-deplete sub-Mpc structures un- axino mass is sufficiently large that the produced gravitinos less they constitute less than 3% of dark matter [32]. are cold. The superpartner spectrum is taken to be identical The right-hand vertical contour that bounds Region II, to that of Fig. 1. the “Sub-Dominant Axino” case, corresponds to Ya˜ < 0.03Y3/2, placing an upper bound on m3/2 which in- is the observed dark matter relic abundance today. Here creases with fa as seen by comparing Fig. 4 and Fig. 5. The lower bound of Region II arises from the adverse Ya˜ and Y3/2 are the thermal axino and gravitino yields arising from scattering and decay processes. Provided effects of relativistic gravitinos arising from axino decay. As discussed in detail in [30], additional relativistic de- that ma˜,m3/2 > keV, both components are cold. Re- gion Ia corresponds to the dark matter energy density grees of freedom can adversely affect matter-radiation dominated by gravitinos, while Region Ib corresponds to equality and are thus constrained by observations of the axino domination. cosmic microwave background, galaxy clustering, and the Lyman-α forest. These bounds apply to axino lifetimes Meanwhile, as the axino mass increases, the lifetime for 13 the decaya ˜ G˜ +a decreases, so the axinos become cos- shorter than 10 sec, and correspond roughly to an addi- mologically→ unstable. Gravitinos from axino decay will tional effective number of neutrino species less than one. then contribute to the total gravitino dark matter abun- Numerically, this implies [30] dance, requiring m3/2 Ya˜ + Y3/2 = ρ0/s0. −5 10 Ta˜ If ma˜ m3/2 the gravitinos produced in axino decay ma˜Ya˜ . 3.4 10 GeV , (17) are highly≫ relativistic, becoming non-relativistic only at × × g∗s(Ta˜) MeV a temperature where Ta˜ is the temperature when the axino decays. Here 3/2 Ta˜ is simply computed by solving for the temperature −2 ma˜ TNR 10 eV . (16) at which H = 1/τ , where τ is defined in Eq. (14). ∼ × GeV a˜ a˜ Plugging in the value of Ya˜ from scattering in Eq. (17) In Region III of Fig. 4 and Fig. 5, the axino mass then yields a substantial limit on TR given by ma˜ & TeV, so that TNR & keV and the gravitinos from axino decay are cold. There is little parameter space for f 2 m 3/2 GeV T < 200 GeV a a˜ this “Fast Decaying Axino” case, since axinos are are no- R 12 × 10 GeV GeV m3/2 longer expected to be the NLSP if ma˜ increases much above a TeV. For m m the above bound on T is much more a˜ ∼ 3/2 R For cosmologically unstable axinos, ma˜ . TeV gives stringent than the one depicted in Fig. 1 from overclo- TNR . keV so that the gravitinos from axino decay sure from gravitinos and axinos. However, by fixing TR 6 to lie on the green contour of Fig. 1, such that gravitino dark matter from thermal production equals the mea- 106 sured value today, then the above inequality for TR can be reinterpreted as a lower bound on ma˜ as a function of m . This constraint is what produces the slanted 3/2 105 lower-right boundary of Region II. 2 In Fig. 6 contours of Ω3/2h =0.11 have been plotted for several values of fa and a fixed superpartner spec- 4 D 10 trum. These are contours of m3/2 Ya˜ + Y3/2 = ρ0/s0 GeV @ R and therefore corresponds to gravitino dark matter in T the “Sub-Dominant Axino” case. The contours become 3 dashed when the values of m3/2 are excluded by the warm 10 dark matter bound Ya˜ < 0.03Y3/2; this corresponds to the rightmost boundary of Region II in Fig. 4 and Fig. 5.
Examining Fig. 6 it is clear that while the angled grav- 102 itino scattering region is further excluded, the vertical gravitino freeze-in region is relatively unaffected. For 11 12 fa = 10 GeV and 10 GeV the axino and gravitino 101 yields in the bound Ya˜ < 0.03Y3/2 are both dominated 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 by the scattering contributions. Thus the TR depen- m32@GeVD dence cancels and the resulting upper bound on m3/2 is dependent only on fa. As fa decreases the m3/2Ya˜ contours move to left as seen in Fig. 2. Therefore the FIG. 6: Gravitino dark matter in Region II, with contours gravitino yield that is to be compared with the axino of m3/2 Ya˜ + Y3/2 = ρ0/s0. Here the {Red, Orange, Yellow, scattering yield becomes dominated by freeze-in, forcing Green} contours correspond to log10 fa/GeV = {9, 10, 11, 12}. The dotted portion of each contour shows values of m3 2 that T to smaller values. Since Y scatt/Y decay m2 T /f 2 / R a˜ 3/2 ∝ 3/2 R a are disallowed by hot dark matter, where hot gravitinos from the upper bound on m3/2 relaxes whenever we cross over axino decay constitute more than 3% of the total abundance. into the freeze-in dominated region. The transition from solid to dashed contour corresponds to Fig. 4 and Fig. 5 depict cosmological restrictions on crossing from Region II into the white region of Fig. 4 and Fig. 5. The superpartner spectrum is taken to be identical to (m3/2,ma˜,fa) which limit the range of collider signals resulting from the decay of the lightest observable-sector that of Fig. 1. superpartner (LOSP)—that is, the lightest R-parity odd superpartner of a standard model particle. Assuming an axion model in which the LOSP couples directly to the sufficiently small that the LOSP decays dominantly to axino (for instance a gluino LOSP in the KSVZ model) gravitinos and has a lifetime directly correlated with the the ratio of LOSP decay rates to axinos and gravitinos is freeze-in production mechanism [23]. For the charged 2 proportional to (m3/2/fa) , then branching ratio to axi- slepton LOSP, decay signals to gravitinos and axinos have 6 been studied in [33], examining the degree to which axino nos (gravitinos) dominates at large (small) m3/2 . Hence and gravitino modes can be distinguished. in Region III at large m3/2 the LOSP lifetime is fixed by fa and the dark matter is produced via axino production. At some intermediate values of m3/2 the LOSP has siz- able branching ratios to both axinos and gravitinos, and B. Cosmological Bounds from Saxions if the axino mass is sufficiently heavy these modes can be distinguished by kinematics. Hence in parts of regions I So far we have ignored the saxion component of the ax- and II it may be possible to measure (m3/2,ma˜,fa). At a ion supermultiplet. With communication between the su- particular value of m3/2, for instance around m3/2 0.3 MeV for squark masses of 500 GeV, the dark matter∼ is persymmetry breaking and PQ sectors mediated at mass scale Λ, the saxion picks up a mass at the same order as produced by gravitino freeze-in. This value of m3/2 is the axino mass, ms F/Λ. This follows from very sim- ilar arguments to those∼ of Section 2—in particular, by inserting one power of Fi of Eq. (9) into the operator of 6 Regardless of the choice of axion model any ultraviolet physics Eq. (7), or more directly from the dimension 6 operator which generates the operator in Eq. (4) will typically induce ax- Φ†Φ X†X/Λ2 in the Kahler potential. In the effective 4 i i ino/goldstino kinetic mixing via R d θ ǫ(A+A†)(X +X†), where theory beneath fa, Planck-scale dynamics induce the op- ∼ ˜ µ † ǫ fa/mPl. In components, ǫGσ ∂µa˜ is removed by shifting erator G˜ → G˜ + ǫa˜, which induces a coupling of the axino to the super- ˜ current of the form ǫa˜J. Therefore, the LOSP typically decays to † † 2 X X(A + A)2 1 a˜ with a branching fraction suppressed by a factor of ǫ relative d4θ m2 ss + h.c. + ...,(18) to the branching fraction to G˜. 2 3/2 Z mPl ≃ 2 7 so that the gravitino mass provides a lower bound for the LSP and axino NLSP. In this case the axino plays an im- saxion mass. portant role in determining viable cosmological histories. The cosmological bounds imposed by saxions have In particular, the overclosure constraint alone provides a been studied in detail in [30]. We find that under the very powerful limit on the reheat temperature, as shown assumption ms >m3/2 these bounds on TR are typically in Figure 6, that can be approximated by less stringent than those coming from axino cosmology. In particular supersymmetric axion models must satisfy 5 fa TR < 3 10 GeV . (20) constraints on × 1012 GeV Saxion overclosure, We identify three very different phases of combined grav- • itino and axino cosmology which we label according to Saxion decays influencing BBN, • the nature of the axino: Relativistic degrees of freedom. • (I) “Cosmologically-Stable Axino” is the case The saxion abundance receives a contribution from scat- where axinos and gravitinos are co-dark matter. tering in the early universe which is identical to the axino scatt scatt (II) “Sub-Dominant Axino” production leads to scattering yield Ys = Ya˜ , as can be seen from the supersymmetric interaction of Eq. (1). We assume a neg- gravitino dark matter from decays, with an upper ligible contribution to the saxion abundance from coher- bound on m3/2 that depends on fa. ent oscillations. Unlike the axino and gravitino, saxion (III) “Fast-Decaying Axino” has a TeV scale mass overclosure does not pose a threat because saxions decay and thus decays sufficiently early that gravitino rapidly; to photons through the electromagnetic analogue dark matter is cold. of Eq. (1) and to axions through the kinetic term of the axion supermultiplet. The regions of gravitino and axino masses that yield Since the saxion will rapidly decay to photons one may these three cosmologies are shown in Fig. 4 and Fig. 5 for 9 12 worry that these photons might ruin the predictions of fa = 10 GeV and 10 GeV. In all three cosmologies, BBN unless the saxion is made heavy enough. However, dark matter may be dominantly produced by gravitino this is not the case, since decays to photons are sup- freeze-in. pressed by a loop factor and so will always be subdomi- The required ranges of TR for cases (II) and (III) are nant to decays to axions shown in Fig. 6 for several values of fa. They give an upper bound on T that is much lower than for the Γ (s aa) 64π2 R → , (19) case of gravitino dark matter without axinos, and is also Γ (s γγ) ∼ α2 → lower than the bound of Eq. (20). Thus the presence of where α is the fine structure constant. While the branch- the axinos increases the likelihood that the production ing fraction of the saxion to axions can in principle be mechanism for gravitino dark matter is freeze-in from de- small, this requires a delicate and unnatural cancellation cays, strengthening the possibility that LHC will provide in the underlying theory. strong evidence for gravitino dark matter by measuring Finally, let us consider the issue of relativistic degrees the LOSP lifetime [23]. Finally, let us note that our con- of freedom. In complete analogy with the axino, the sax- clusions are unaltered by cosmological considerations of ion will decay and produce relativistic energy in axions the saxion for any value of its mass greater than m3/2. which is subject to the bound in Eq. (17) taken from [30] except with m Y and T replaced with m Y and T . a˜ a˜ a˜ s s s Acknowledgments Plugging in for Ys, the saxion yield from thermal scat- tering (which is essentially equal to the axino yield from thermal scattering), and plugging in for Ts, the tempera- We thank Mina Arvanitaki, Kiwoon Choi, and Peter ture at the time of saxion decay where τ −1 m3/64πf 2, Graham for useful discussions. This work was supported s ∼ s a one acquires a bound on TR given by in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of 1/2 8 fa ms Energy under Contract DE-AC02-05CH11231 and by the TR < 5 10 GeV 12 , × × 10 GeV GeV National Science Foundation on grant PHY-0457315. which is generally weaker than the TR bound from over- closure depicted in Fig. 1. Appendix A: A Simple Theory
V. CONCLUSIONS Consider a minimal theory of PQ and supersymmetry breaking defined by If supersymmetry and the QCD axion coexist, then a = d4θK + d2θ W + h.c., (A1) large range of parameter space will include a gravitino L Z Z 8 where the Kahler potential and and superpotential are where the superscripts denote derivatives with respect to defined by the function arguments. Plugging these expressions into the action, canonically normalizing the fermion kinetic Φ†ΦX terms, and then diagonalizing the fermion mass matrix, K = G(Φ†Φ) + H(X†,X)+ + h.c. (A2) Λ one discovers that one linear combination of fermions W = FX. (A3) from Φ and X is massless, as is expected of the goldstino. Meanwhile, the orthogonal axino component acquires a Here Φ and X are the PQ and supersymmetry breaking mass fields, respectively, and G and H are as of yet unspec- ified real functions. Since Φ is PQ charged, it cannot F 1 1 be present in the superpotential. Also, note that G is ma˜ = (1) 2 (2) (1,1) + ..., (A5) † 2Λ G + f G H purely a function of Φ Φ so that the PQ symmetry is h i a h i h i left unbroken. We interpret the 1/Λ suppressed operator coupling Φ and X as something generated by unspecified where the ellipses denote terms higher order in Λ. Hence, high-scale dynamics. we confirm the result of the simple operator argument in Demanding that G and H have a form such that Φ = Eq. (5). h i fa and X = 0 at the minimum of the scalar potential impliesh thati
G(1) = f 2 G(2) + f 4 G(3) (A4) h i a h i a h i f 2 H(1,2) = H(2,1) = a , h i h i −Λ3( G(1) + f 2 G(2) )2 h i a h i
[1] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979) ; M. A. [hep-ph/9503233] Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. [17] H. Pagels, J. R. Primack, Phys. Rev. Lett. 48, 223 Phys. B166, 493 (1980). (1982). S. Weinberg, Phys. Rev. Lett. 48, 1303 (1982). [2] M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. M. Y. Khlopov, A. D. Linde, Phys. Lett. B138, 265-268 B 104, 199 (1981); A.R. Zhitnitsky, A. R. Sov. J. Nucl. (1984). D. V. Nanopoulos, K. A. Olive, M. Srednicki, Phys. 31, 260 (1980). Phys. Lett. B127, 30 (1983). T. Moroi, H. Murayama, [3] J. Wess and J. Bagger, Supersymmetry and Supergrav- M. Yamaguchi, Phys. Lett. B303, 289-294 (1993). ity, (Princeton University Press, Princeton, New Jersey, [18] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 1992); D. Balin and A. Love, Supersymmetric Gauge (1986). Field Theory and String Theory, (Taylor & Francis [19] Y. Grossman, T. Kashti, Y. Nir, E. Roulet, JHEP 0411 Group, New York, 1994). (2004) 080. [hep-ph/0407063]. [4] T. Higaki, R. Kitano, [arXiv:1104.0170 [hep-ph]] [20] L. J. Hall, J. March-Russell, S. West, [arXiv:1010.0245 [5] M. Kamionkowski and J. March-Russell, Phys. Lett. B [hep-ph]] 282, 137 (1992) [arXiv:hep-th/9202003]. [21] C. Cheung, L. J. Hall, D. Pinner, [6] R. Kallosh, A. D. Linde, D. A. Linde and L. Susskind, [arXiv:1103.3520 [hep-ph]]. Phys. Rev. D 52, 912 (1995) [arXiv:hep-th/9502069]. [22] L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, [7] R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, JHEP 1003, 080 (2010) [arXiv:0911.1120 [hep-ph]] R. Watkins and L. M. Widrow, Phys. Lett. B 282, 132 [23] C. Cheung, G. Elor, L. J. Hall, [arXiv:1103.4394 [hep- (1992) [arXiv:hep-ph/9203206]. ph]]. [8] H. M. Georgi, L. J. Hall and M. B. Wise, Nucl. Phys. B [24] C. Cheung, G. Elor, L. J. Hall, P. Kumar, 192, 409 (1981). [arXiv:1010.0022 [hep-ph]]. [9] K. S. Babu, I. Gogoladze and K. Wang, Phys. Lett. B [25] C. Cheung, G. Elor, L. J. Hall, P. Kumar, 560, 214 (2003) [arXiv:hep-ph/0212339]. [arXiv:1010.0024 [hep-ph]]. [10] A. G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D [26] C. Cheung, Y. Nomura, J. Thaler, JHEP 1003, 073 69, 015007 (2004) [arXiv:hep-ph/0210172]. (2010). [arXiv:1002.1967 [hep-ph]]. [11] C. Cheung, JHEP 1006, 074 (2010) [arXiv:1003.0941 [27] C. Cheung, J. Mardon, Y. Nomura, J. Thaler, JHEP [hep-ph]]. 1007, 035 (2010). [arXiv:1004.4637 [hep-ph]]. [12] L. Randall, R. Sundrum, Nucl. Phys. B557, 79-118 [28] T. Goto, M. Yamaguchi Phys. Lett. bf B276, 103-107 (1999) [arXiv:hep-th/9810155] (1992). [13] G. F. Giudice, M. A. Luty, H. Murayama, R. Rattazzi, [29] A. Strumia, JHEP 1006, 036 (2010). [arXiv:1003.5847 JHEP 12, 027, (1998) [arXiv:hep-ph/9810442] [hep-ph]]. [14] N. Abe, T. Moroi, M. Yamaguchi, JHEP 01, 010, [30] M. Kawasaki, K. Nakayama, M. Senami JCAP 0803 (2002) [arXiv:hep-ph/0111155] 009 (2008) [arXiv:0711.3083 [hep-[ph]] [15] K. Choi, K. S. Jeong, W. -I. Park, C. S. Shin, JCAP [31] E. J. Chun, H. B. Kim and J. E. Kim, Phys. Rev. 0911, 018 (2009). [arXiv:0908.2154 [hep-ph]]. Lett. 72, 1956 (1994) H. B. Kim and J. E. Kim, [16] E. J. Chun, A. Lukas, Phys. Lett. B357, 43-50 (1995). Nucl. Phys. B 433, 421 (1995) L. Covi, J. E. Kim 9
and L. Roszkowski, Phys. Rev. Lett. 82 (1999) 4180 and A. Riotto, Phys. Rev. D 71, 063534 (2005); C. L. [hep-ph/9905212]; L. Covi, H. B. Kim, J. E. Kim and L. Bennett, M. Bay, M. Halpern, G. Hinshaw, C. Jack- Roszkowski, JHEP 0105 (2001) 033 [hep-ph/0101009]; son, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, L. Covi, L. Roszkowski and M. Small, JHEP 0207 L. Page, et al., Astrophys. J. 583, 1 (2003); U. Seljak, (2002) 023 [hep-ph/0206119]; L. Covi, L. Roszkowski, A. Slosar, and P. McDonald, JCAP 0610 (2006) 014 R. Ruiz de Austri and M. Small, JHEP 0406 (2004) 003 [astro-ph/0604335v4] [hep-ph/0402240]; A. Brandenburg, L. Covi, K. Ham- [33] A. Brandenburg, L. Covi, K. Hamaguchi, L. Roszkowski, aguchi, L. Roszkowski and F. D. Steffen, Phys. Lett. B F. D. Steffen, Phys. Lett. B617, 99-111 (2005). 617 (2005) 99 [hep-ph/0501287] [hep-ph/0501287]. [32] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese,