Supersymmetric inflation and baryogenesis via Extra-Flat directions of the MSSM
aa r X i v : . [ h e p - ph ] J u l IFT-UAM/CSIC-08-04
Supersymmetric inflation and baryogenesis via Extra-Flatdirections of the MSSM
John McDonald ∗ Cosmology and Astroparticle Physics Group,University of Lancaster, Lancaster LA1 4YB, United Kingdom
Osamu Seto † Instituto de F´ısica Te´orica UAM/CSIC,Universidad Aut´onoma de Madrid, Cantoblanco, Madrid 28049, Spain
Abstract
One interpretation of proton stability is that it implies the existence of extra-flat directions of theminimal supersymmetric standard model, in particular u c u c d c e c and QQQL , where the operatorslifting the potential are suppressed by a mass scale Λ which is much larger than the Planck mass,Λ > ∼ GeV. Using D -term hybrid inflation as an example, we show that such flat directionscan serve as the inflaton in supersymmetric inflation models. The resulting model is a minimalversion of D -term inflation which requires the smallest number of additional fields. In the casewhere Q -balls form from the extra-flat direction condensate after inflation, successful Affleck-Dinebaryogenesis is possible if the suppression mass scale is > ∼ − GeV. In this case thereheating temperature from Q -ball decay is in the range 3 −
100 GeV, while observable baryonisocurvature perturbations and non-thermal dark matter are possible. In the case of extra-flatdirections with a large t squark component, there is no Q -ball formation and reheating is viaconventional condensate decay. In this case the reheating temperature is in the range 1 −
100 TeV,naturally evading thermal gravitino overproduction while allowing sphaleron erasure of any large B − L asymmetry. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Successful models of supersymmetric (SUSY) inflation should ideally satisfy a numberof requirements: natural compatibility with supergravity (SUGRA), lack of fine-tuned cou-plings, successful post-inflation era including reheating and baryogenesis, and compatibilitywith unified models of particle physics. With respect to these conditions, SUSY hybridinflation models have a particular attraction [1, 2]. They can achieve sufficient inflationwithout requiring very small or fine-tuned couplings, and in the case of D -term hybrid in-flation they are naturally compatible with SUGRA [3]. Focusing on the D -term inflationcase, a natural question is the origin of the fields in the D -term inflation sector. The U (1)gauge field and charged vector pair Φ ± of D -term inflation might be understood as compo-nents of an extended gauge theory. However, the inflaton is usually a gauge singlet whichis added to the model for no other reason . If we do not add such a singlet, can D -termhybrid inflation still occur? Here we argue that it can. The vector pair will naturally coupleto any gauge-invariant combination of fields in the MSSM. Such gauge-invariant products(monomials) also characterise flat directions of the MSSM. Thus a natural possibility is thata flat direction can play the role of the inflaton in D -term inflation models . In this modelthe number of additional fields required for inflation is reduced to just a U (1) gauge fieldand the Φ ± vector pair, so providing a minimal version of D -term inflation. As we willshow, conventional MSSM flat directions lifted by Planck scale-suppressed gauge-invariantsuperpotential terms are unsuitable. This is because such terms generally lift the flat di-rection scalar at field strengths well below the value required for inflation. However, it isknown that certain gauge-invariant superpotential terms must be suppressed by more thanthe Planck scale or forbidden entirely. The d = 4 operators u c u c d c e c and QQQL will lead torapid proton decay if they are only Planck scale-suppressed [9]. One way this problem can besolved is by assuming that the underlying complete theory introduces a dynamical suppres-sion factor into the non-renormalisable superpotential interactions, such that the effectivemass scale suppressing the dangerous operators is Λ > ∼ GeV [9]. It is also possible that Models exist which attempt to identify the inflaton with a known field, such as a right-handed sneutrino[4, 5, 6]. An interesting model using MSSM flat directions as inflatons, which has a quite different philosophy withrespect to fine-tunings, is given in [7]. See also [8]. u c u c d c e c ) and ( QQQL ) . We will refer to a flat directionfor which this is true as an ‘extra-flat direction’. An alternative interpretation of the absenceof proton decay is in terms of a discrete symmetry which eliminates the dangerous d = 4 op-erators [10]. In this case it is possible that the higher-order operators will be unsuppressed.However, as we will show, such unsuppressed flat directions, even if higher-order, cannotserve as an inflaton. If the existence of extra-flat directions is the correct interpretation ofthe absence of proton decay in the MSSM, then an extra-flat direction scalar could serveas the inflaton in a D -term inflation model. The extra-flat direction potential at large fieldvalues is naturally lifted to an inflationary plateau by its gauge-invariant superpotentialcoupling to Φ + Φ − . Reheating and possibly baryogenesis would then come from the decay ofthe flat direction inflaton, via either Q -ball decay or conventional homogeneous condensatedecay, depending on the t squark component of the flat direction. In this paper we willstudy D -term inflation along an extra-flat direction of the MSSM. The paper is organisedas follows. In Section 2 we discuss extra-flat directions and the resulting D -term inflationmodel. In Section 3 we discuss reheating and baryogenesis. In Section 4 we present ourconclusions. II. D -TERM INFLATON ALONG EXTRA-FLAT DIRECTIONSA. Potential We consider a flat direction Φ in the MSSM and introduce two additional fields Φ ± charged under a U(1) gauge group with the Fayet-Illiopoulos term ξ . The superpotential is W = λ Φ m mM m − + λ Φ n nM n − Φ + Φ − , (1)3here λ , are Yukawa couplings and M is the reduced Planck mass, M = M P l / √ π . Wewill present results for general m and n , specialising to the case of most interest m = n = 4,corresponding to Φ ∼ u c u c d c e c or QQQL . Proton stability in the case of u c u c d c e c or QQQL requires that λ < ∼ − , corresponding to an effective suppression mass scale Λ = M/λ > ∼ GeV. However, λ is unconstrained by phenomenology and will be determinedby the inflation model. The scalar potential in the global SUSY limit is then V = (cid:12)(cid:12)(cid:12)(cid:12) λ φ n nM n − (cid:12)(cid:12)(cid:12)(cid:12) ( | φ + | + | φ − | ) + (cid:12)(cid:12)(cid:12)(cid:12) λ φ m − M m − + λ φ n − M n − φ + φ − (cid:12)(cid:12)(cid:12)(cid:12) + g ξ + | φ + | − | φ − | ) . (2)The supersymmetric global minimum is located at ( φ, φ + , | φ − | ) = (0 , , p ξ ) . (3)If (cid:12)(cid:12)(cid:12)(cid:12) λ λ φ m + n − M m + n − (cid:12)(cid:12)(cid:12)(cid:12) ≪ ( g ξ ) (4)is satisfied, the mixing between φ + and φ − is negligible. The potential is then simplified to V ≃ (cid:12)(cid:12)(cid:12)(cid:12) λ φ n nM n − (cid:12)(cid:12)(cid:12)(cid:12) ( | φ + | + | φ − | ) + (cid:12)(cid:12)(cid:12)(cid:12) λ φ m − M m − (cid:12)(cid:12)(cid:12)(cid:12) + g ξ + | φ + | − | φ − | ) . (5)The critical value of φ is given by | φ c | ≡ nM n − p g ξ | λ | ! /n , (6)which determines the stability of the φ − field at the origin. The origin is a false vacuum for | φ | > | φ c | , while it is unstable for | φ | < | φ c | . A SUSY mass term W ⊃ µ Φ + Φ − has not been included. This term would induce n minima withnonvanishing VEV for Φ, which consist of squark and/or slepton VEV and lead to large baryon or leptonnumber violation in the MSSM. Although in most cases there is no symmetry which can exclude such aterm, we note that for the case m = n this term can be excluded by an R-symmetry which allows theterms Φ n and Φ n Φ + Φ − . In addition, if µ is less than the scale of soft SUSY breaking terms, the minimumof the potential can be at Φ = 0, while for larger µ there can be directions in the complex Φ plane alongwhich the field evolution can avoid the minima with Φ = 0. Note that there is a SUSY flat direction when µ = 0 and φ = 0, such that | φ + | − | φ − | = ξ . However,the minimum with φ + = 0 is selected since φ + gains a large mass when φ = 0 during inflation. . Inflationary expansion For | φ | > | φ c | , φ − = 0 is a local minimum and there is the false vacuum energy from the D -term, which drives inflation. The potential during inflation is given as V ≃ g ξ (cid:18) g π ln σ n Λ n ∗ (cid:19) , (7)where σ = √ Re ( φ ) is the canonically normalised inflaton and Λ ∗ is the renormalisationscale. Inflation ends when the inflaton reaches the larger of σ c ≡ √ | φ rmc | and σ f ≡ √ ngM π , (8)where σ f corresponds to the end of slow-roll. However, a non-vanishing F-term potential isalso present in this model. Hence, we need to ensure that the condition V F ≪ V D is satisfied,which requires that (cid:12)(cid:12)(cid:12)(cid:12) λ φ m − M m − (cid:12)(cid:12)(cid:12)(cid:12) ≪ g ξ (9)is satisfied. Note that when this is satisfied, equation (4) is also satisfied. The dynamics ofthe inflaton field is similar to that in the minimal D -term hybrid inflation model [1]. Thesolution of the slow-roll field equations is σ ( N ) = σ + ng N M π . (10)Here, σ = max[ σ f , σ c ] is the expectation value of inflaton when inflation terminates. Thespectral index is n s = 1 − N (cid:18) π σ ng M N (cid:19) − , (11)while the value of ξ / normalised to the curvature perturbation P ζ is ξ / M = (cid:18) nP ζ N (cid:19) / (cid:18) π σ ng M N (cid:19) − / . (12)In the case of σ = σ c , which we will show is true in examples of interest, we find2 π σ ng M N = (cid:18) λ c λ (cid:19) /n (13)with λ c = (cid:18) π ng N M (cid:19) n/ (cid:0) g ξn M n − (cid:1) / . (14)5 = λ c corresponds to σ N = 2 σ c , with σ ≈ σ c throughout inflation when λ < λ c . When λ ≫ λ c as well as the case of σ = σ c , the spectral index is n s = 1 − /N ≈ .
98, as inconventional D -term inflation, while the value of ξ / required to account for the observedcurvature perturbation ( P / ζ = 4 . × − ) is ξ / = 7 . × n / GeV. On the other hand,in the case where λ ≪ λ c , the spectral index approaches n s = 1 while the value of ξ / isreduced by a factor ( λ /λ c ) / n . C. Comparison with observations
The spectral index observed by WMAP, n s = 0 . ± .
016 (1- σ ) [11], is substantiallysmaller the D -term inflation value. In addition, WMAP data permits at most an O(10)%contribution to the CMB power spectrum from cosmic strings [12, 13, 14], which impliesthat ξ / < ∼ × GeV. (Here we have used Gµ = 2 × − for the l = 10 WMAPnormalised string tension [15].) One way to interpret the WMAP observations is that theycorrespond to an adiabatic curvature perturbation with n s ≈ λ sufficiently small comparedwith λ c . In this case the apparent spectral index of the combined perturbation is effectivelylowered and can be in agreement with the 3-year WMAP data analysis [16]. It is a strikingfeature of D -term inflation models in general that they have a solution which increases n s while decreasing the cosmic string contribution, just as required for this interpretation of theWMAP observations. With respect to this possibility, the extra-flat direction model has apossible advantage over conventional D -term inflation. The contribution of cosmic strings tothe CMB power spectrum is proportional to µ = (2 πξ ) . In the case of conventional D -terminflation with n s = 1, the value of ξ in the limit λ ≪ λ c is proportional to λ . Therefore λ must lie within a rather narrow range of values for the cosmic string contribution to beO(10)%. In the case of the extra-flat direction inflaton, the dependence is ∝ λ /n . Thereforethe CMB contribution varies much more gradually with λ e.g. ξ ∝ λ / for the case n = 4.Thus an O(10)% contribution is obtained for a much wider range of λ , making it perhapsa more natural possibility than in conventional D -term inflation.6 . Constraints from cosmic string bound, SUGRA and potential flatness We first check that the 10% cosmic string condition ξ / ≈ × GeV can be satisfied forreasonable values of g when | φ c | is small enough compared with M for SUGRA correctionsto be neglected. We will require that | φ c | < kM , with k < ∼ .
3, so that | φ c | < ∼ . M . Fromequation (6), this implies that g < ∼ λ k n Mnξ / . (15)For the case n = 4 and ξ / ≈ × GeV, equation (15) implies that g < ∼ . λ (cid:18) k . (cid:19) . (16)Thus λ should not be small compared with 1 if g is not very small compared with 1. Fromequation (12), to suppress ξ / from 7 . × GeV to 4 × GeV in the case n = 4 werequire that λ c /λ ≈ λ ≈ × − g − . (17)Equations (16) and (17) imply that g < ∼ . (cid:18) k . (cid:19) (18)and λ > ∼ . (cid:18) . k (cid:19) . (19)Thus k > ∼ . λ < ∼ . < ∼ k < ∼ . g ≈ . − .
03 and 0 . < ∼ λ < ∼
1. Therefore, as in conventional D -terminflation in the small coupling limit, g must be somewhat smaller than the Standard Modelgauge couplings [13]. In addition, λ must be much larger than λ . We next evaluateEq. (9) to find the condition on λ for F-term corrections not to spoil the flatness of theinflaton potential. In general we find λ ≪ √ gξk m − M . (20) We have assumed that σ c > σ f . For the case n = 4 this requires that ξ / /M > | λ | g /π . With ξ / ≈ × GeV and g ≈ .
02 this is easily satisfied. m = 4 this gives λ ≪ × − g (cid:18) . k (cid:19) (cid:18) ξ / × GeV (cid:19) . (21)Thus for values of λ which satisfy the proton decay constraint, λ < ∼ − , the flat directionpotential is easily sufficiently flat to serve as an inflaton. However, for the case of unsup-pressed n = m = 4 flat directions with λ ∼
1, the F-term would violate the flatness of theflat-direction inflaton potential. An alternative solution of the proton decay problem is toconsider elimination of the m = n = 4 operators entirely by a symmetry. In this case weexpect to have unsuppressed operators with m = n = 8, such that λ ∼ λ ∼
1. However,in this case the F-flatness condition will still be violated. For n = 8, equation (15) impliesthat g < ∼ × − λ (cid:18) k . (cid:19) . (22)To suppress ξ / to 4 × GeV, with n = 8 we need λ c /λ ≈ . × . This implies that λ = 9 × − g − . (23)Therefore if λ < ∼ g > ∼ .
03. Equation (23) combined with equation (22) gives g < ∼ . (cid:18) k . (cid:19) . (24)Thus k > ∼ . g > ∼ .
03. The F-term flatness condition equation (20) for m = 8is λ < ∼ × − (cid:16) g . (cid:17) (cid:18) . k (cid:19) (cid:18) ξ / × GeV (cid:19) . (25)Thus the m = n = 8 flat direction will also need to be extra-suppressed to have a flatinflaton potential, even if the m = n = 4 term is completely eliminated by a discretesymmetry. Therefore extra-flat directions are essential for an MSSM flat direction to playthe role of the inflaton in D -term inflation. E. Post-inflationary evolution
Including soft SUSY breaking terms, the potential is V = m φ | φ | + Am / λ φ m mM m − + H . c . + g ξ | φ − | (cid:12)(cid:12)(cid:12)(cid:12) φφ c (cid:12)(cid:12)(cid:12)(cid:12) n + (cid:12)(cid:12)(cid:12)(cid:12) λ φ m − M m − (cid:12)(cid:12)(cid:12)(cid:12) + g ξ − | φ − | ) , (26)8here we used equation (6). Here m φ is the soft SUSY breaking scalar mass. The potentialhas two SUSY non-renormalisable terms: the third term g ξ ( | φ − | /ξ ) | φ n /φ n c | and thefourth term | λ φ m /M m − | . Assuming that m − < n (since the case of most interest willbe that where m = n ), the third term is dominant if φ & φ ∗ , where φ n − m +1 ∗ ≡ λ φ n c M m − p g ξ (cid:18) ξ | φ − | (cid:19) / , (27)while the fourth term is dominant if φ < φ ∗ . An important point in what follows is thatfor φ > ∼ φ ∗ , the A -term will be effectively suppressed compared with the usual case of anMSSM flat direction with potential stabilised by a non-renormalisable term. This is becausethe A -term is coming from the first term in the superpotential, equation (1), whereas thenon-renormalisable term in the scalar potential is from the second term. As a result, thebaryon asymmetry generated by the Affleck-Dine mechanism [17, 18] will be suppressedrelative to the MSSM flat direction case. Once inflation ends, the φ − field oscillates aroundthe minimum h φ − i = √ ξ . The φ field will oscillate around the origin dominated by eitherthe | λ φ m − /M m − | or the |h φ − i| g ξ | φ n /φ n c | term, depending on the amplitude. Whilethe amplitude of the φ oscillation is large, the energy density of φ will decrease more rapidlythan that of φ − ( V ∝ φ d implies that ρ ∝ a − d/ ( d +2) , with d ≥ φ oscillations and d = 2for φ − oscillations), so the Universe initially becomes φ − dominated. In the following we willassume that the φ − oscillations efficiently decay into radiation. (We will comment on howour results are altered if this is not satisfied.) Due to the φ − decay, the radiation producestwo distinct thermal corrections to the potential equation (26). The φ field is expected toacquire a thermal mass term h T | φ | , (28)with h being a coupling between φ and a particle in the thermal bath [19] in the case wherethe expectation value of the field is relatively small and the radiation temperature is highenough, and also a logarithmic term αT ln | φ | T , (29)which appears at the two-loop level through the running of couplings with non-vanishing φ [20]. Here, α is a constant of order of 10 − and its sign can be positive or negative. For In general, the minimum of the potential is at | φ − | = ξ / (cid:0) − | φ/φ c | n (cid:1) / . This rapidly tends to | φ − | = ξ / as the φ oscillations are damped from φ c to small amplitudes. α is negative. (For a positive α , the fieldoscillates around the origin by either the mass m φ , the thermal mass or this two-loop effectand simply decays into radiation.) The potential with the two-loop induced logarithmicpotential is V ( φ ) = m φ (cid:18) K ln | φ | Λ (cid:19) | φ | + Am / λ φ m mM m − + H . c . + αT ln | φ | T + g ξ | φ − | (cid:12)(cid:12)(cid:12)(cid:12) φ n φ n c (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) λ φ m − M m − (cid:12)(cid:12)(cid:12)(cid:12) , (30)where we include the radiative correction to m φ with the direction dependent coefficient K .For MSSM flat directions which do not include a large top quark component, K ≃ − − [21, 22]. However, a large top squark component can drive K to positive values.A negative K is the source of the spatial instability which leads to Q -ball formation inthe gravity mediated SUSY breaking model [21]. For a negative α , as shown in Ref. [23],the thermal mass term cannot appear because of a relatively large expectation value ofthe field. Here, the φ field is trapped with nonvanishing value by the thermal logarithmicterm, equation (29), and the non-renormalizable term, until the temperature decreases to acertain value. As the temperature falls, the expectation value of φ becomes small. Whenthe φ becomes as small as | φ os | ≃ ( − α ) T m φ , (31) φ starts to oscillate around the origin with the angular momentum in the φ space inducedby A -term, which is equivalent to the charge density (baryonic and/or leptonic) carried by φ [17]. Provided that the reheating by the φ − decay is completed before φ starts to oscillate,we find from equation (31) that ρ φ ρ R (cid:12)(cid:12)(cid:12)(cid:12) t os ≃ − α ) π g ∗ . (32)Here, g ∗ is the effective total degrees of freedom of the relativistic species in the radiation.Since the ratio in equation (32) is of order of 10 − , the φ field oscillations (or the Q -balldensity formed from the φ condensate if K <
0) soon dominates the Universe.
III. REHEATING AND BARYOGENESIS
Reheating in this model is from the decay of the extra-flat direction inflaton field. Thereheating temperature will therefore depend on whether or not the flat direction condensate10ragments into Q -balls, which in turn depends on the t squark content of the flat direction[22]. A. Q -ball formation and decay The φ field oscillates around the origin coherently to begin with, but there is a spatialinstability of its fluctuations due to the negative K . After inhomogeneities in the field grow,the coherent φ fragments and, as a result, Q -balls are formed [21, 24]. Here we brieflysummarize properties of Q -balls in gravity mediated SUSY breaking models. The radius ofa Q -ball, R , is estimated as R ≃ / ( | K | m φ ) [21]. By numerical calculations, it was shownthat almost all the produced charge is stored inside Q -balls, and that a good fit to the Q -ballcharge is Q ≃ ¯ β (cid:18) | φ os | m φ (cid:19) ǫ Q (33)with ǫ Q = ǫ for ǫ & ǫ c ǫ c for ǫ < ǫ c , (34)and ǫ ≡ n q n φ (cid:12)(cid:12)(cid:12)(cid:12) t os ≃ q | A | (cid:18) m / m φ (cid:19) sin δ × Min (cid:20)(cid:18) λ λ ∗ (cid:19) , (cid:21) (cid:18) m φ H os (cid:19) , (35)where δ is the CP violating phase, ǫ c ≃ − and ¯ β = 6 × − [25]. (For ǫ < ǫ c thecondensate will fragment to pairs of oppositely charged Q -balls.) The last two factors inequation (35) are, respectively, the suppression of the baryon asymmetry due to the effectivesuppression of the A -term relative to the non-renormalisable term once φ > ∼ φ ∗ (where λ ∗ isdefined below), and the enhancement due to H os ≪ m φ at the onset of φ oscillations, whichallows the B violating A -term to act over many φ oscillations before expansion diminishesthe A -term .The decay temperature of Q -ball is given by [26] T d ≃ p f s (cid:16) m φ (cid:17) / (cid:18) Q (cid:19) / GeV , (36) In the case where the φ − field does not decay efficiently to radiation, the Universe after inflation willbe dominated by φ − oscillations and onset of oscillations will be typically determined by an order H correction to the φ mass squared due to non-minimal K¨ahler interactions of the form | φ − | | φ | . In thiscase m φ ≈ H os in Eq. (35). & f s ≥ Q -balls can decay into finalstates consisting purely of scalar particles. Since Q -balls come to dominate the Universe inour scenario, the decay temperature gives the reheating temperature at the onset of radiationdominated Universe. The resultant emitted charge to entropy ratio is given by n q s = 34 T d m φ ǫ. (37) B. Affleck-Dine baryogenesis
The baryon asymmetry is generated by the B and CP violating A -term when φ starts tooscillate around the origin [17]. As usual in D -term inflation, there is no order H correction tothe A -term before φ starts oscillating, since φ + = 0 throughout [27]. In addition, in the caseof a non-singlet inflaton there can be no linear coupling of the inflaton I to superpotentialmonomials W in the K¨ahler potential of the form I † W , which would generate an order HA -term correction [18]. Therefore the phase of the inflaton relative to the A -term at theonset of φ oscillations, θ , is determined by its initial random value during inflation, in whichcase sin δ ≈ (sin 2 θ ) /
2. This phase gives the CP violating phase required for Affleck-Dinebaryogenesis, with n B ∝ θ for θ small compared with 1. In the estimation of the resultantbaryon asymmetry produced by the Affleck-Dine mechanism, the important quantity is theamplitude of the AD field when it starts to oscillate, φ os . For φ os > ∼ φ ∗ , the amplitude isgiven by m φ ≃ ng ξ (cid:12)(cid:12)(cid:12)(cid:12) φ n − φ n c (cid:12)(cid:12)(cid:12)(cid:12) , (38)where we assume | φ − | = ξ . On the other hand, for φ os < ∼ φ ∗ , the amplitude is given by m φ ≃ ( m − (cid:12)(cid:12)(cid:12)(cid:12) λ φ m − M m − (cid:12)(cid:12)(cid:12)(cid:12) . (39)The former applies in the case of a small λ < ∼ λ ∗ , with λ ( n − ∗ ≡ p g ξ | φ c | n ! m − (cid:18) m φ n (cid:19) n − m +12 M ( m − n − , (40)while the latter corresponds to a large λ > ∼ λ ∗ . For the case λ < ∼ λ ∗ we obtain | φ os | m φ = ng ξ | φ c | n m n − φ ! n − . (41)12quation (33) then gives Q ≃ ¯ β ng ξ | φ c | n m n − φ ! n − ǫ Q . (42)The expansion rate at the onset of φ oscillations during radiation domination can beobtained from equations (31) and (41), H os m φ = (cid:18) π g ∗ α (cid:19) / | φ os | M , (43)with | φ os | M = (cid:18) √ nm φ λ ξ / (cid:19) n − . (44)For n = 4 this gives, H os m φ ≈ × − α − / λ − / (cid:16) m φ (cid:17) / (cid:18) × GeV ξ / (cid:19) / . (45)Therefore H os /m φ ≈ − . This justifies neglect of H corrections to the soft SUSY breakingterms at the onset of φ oscillations.For n = 4, equation (42) becomes Q ≃ . × (cid:18) . g (cid:19) / (cid:18) × GeV ξ / (cid:19) / (cid:18) | φ c | . M (cid:19) / (cid:18) m φ (cid:19) / ǫ Q . (46)Then from equations (36), (37) and (46), and using m φ /H os ≈ , the decay temperatureand baryon asymmetry are given by T d ≃ p f s (cid:18) . √ ǫ Q (cid:19) (cid:16) m φ (cid:17) / (cid:18) . g (cid:19) − / (cid:18) × GeV ξ / (cid:19) − / (cid:18) | φ c | . M (cid:19) − / GeV , (47)and n q s ≃ × − (cid:18) . √ ǫ Q (cid:19) (cid:16) ǫ − (cid:17) p f s (cid:16) m φ (cid:17) / (cid:18) . g (cid:19) − / (cid:18) × GeV ξ / (cid:19) − / (cid:18) | φ c | . M (cid:19) − / . (48)The observed baryon asymmetry is n q /s = (1 . ± . × − . Hence ǫ < ∼ − is necessaryto account for the observed B asymmetry. The Q -ball decay temperature, which gives thereheating temperature, is in the range 3-100 GeV for 1 ≤ f s < ∼ . For λ < ∼ λ ∗ , fromequation (35) we have ǫ ≈ (0 . − λ /λ ∗ )( m φ /H os ) θ . The random phase of φ duringinflation would be expected to be of order 1; therefore in order to generate the observedbaryon asymmetry we require that λ ≈ (10 − − − ) λ ∗ . With m = n = 4, λ ∗ is, λ ∗ ≃ . × − (cid:16) m φ (cid:17) / (cid:18) × GeV ξ / (cid:19) − / λ / . (49)13ince λ should not be very small if g is not very small, λ ∗ ≈ − is likely. Thereforeto account for the observed baryon asymmetry with θ ≈ λ < ∼ − . Thiscorresponds to suppression of the QQQL or u c u c d c e c superpotential terms by a mass scaleΛ > ∼ GeV. Thus a much larger suppression is necessary for successful baryogenesisthan is required by proton stability. Note that it may be possible for Q -balls to decay ata temperature greater than that of the electroweak transition if f s ≈ , corresponding to Q -ball decay to purely scalar final states. In this case any dangerous baryon asymmetry willbe erased by B + L violating sphaleron fluctuations. C. Baryon isocurvature perturbations
The CP violating phase δ is given by the phase of φ during inflation relative to the A -term, which defines the real direction. Therefore in the case where reheating is via Q -ball decay, ǫ ≈ ( λ /λ ∗ ) θ implies that θ ≈ − ( λ ∗ /λ ) in order to have ǫ ≈ − , asrequired for successful baryogenesis with f s = 1. Quantum fluctuations of φ in the phasedirection will lead to baryon isocurvature perturbations, which can be large when θ ≪ α BI , where [28, 29] α BI = (cid:18) Ω B Ω DM (cid:19) f θ H π P R φ , (50)with δn B /n B ≈ f θ δθ . In our case f θ ≈ /θ . The present observational limit is α BI < . θ ≈ − ( λ ∗ /λ ), Ω DM = 0 .
23 and Ω B = 0 .
04, this gives a upper bound on
H/φ , H πφ < ∼ − (cid:18) λ ∗ λ (cid:19) . (51) ξ / ≈ × GeV, corresponding to O(10)% cosmic strings, implies that H = 2 . × g GeV. Since we are considering φ ≈ φ c ≈ (0 . − . M , we therefore have H πφ ≈ (0 . − . × − g , (52)Thus with g ≈ . − .
03, the baryon isocurvature perturbation is sufficiently small if λ /λ ∗ < ∼ − . The correct baryon asymmetry then requires that θ > ∼ − . Thus even if theinitial random phase of the flat direction field could satisfy θ ≪
1, the flat direction wouldstill have to be suppressed by Λ > ∼ GeV in order to avoid large baryon isocurvature14erturbations. For Λ ≈ GeV and θ ≈ − , the correct baryon asymmetry will begenerated together with a potentially observable baryon isocurvature perturbation. D. Non-thermal dark matter
The reheating temperature is ≈ Q -ball decay to purely scalarfinal states is kinematically suppressed, such that f s = 1. This low reheating temperatureimplies that Q -balls may decay below the freeze-out temperature of neutralino LSPs, inwhich case Q -ball decay will also produce non-thermal LSP dark matter particles. In factdark matter particles are often overproduced, in particular for the standard bino-like neu-tralino LSP. Although several ways to avoid this problem have been proposed by taking analternative choice of the LSP [23, 31, 32, 33], perhaps the simplest ones are to assume aHiggsino-like neutralino LSP [32], or a gravitino LSP [23] with a sneutrino NLSP to escapeBBN constraints [34]. E. Reheating from flat direction condensate decay without Q -ball formation In the case where the inflaton corresponds to a flat direction with a large t squark compo-nent, the φ condensate will not fragment to Q -balls since K > B − L conserving u c u c d c e c and QQQL directions, the baryonasymmetry from Affleck-Dine baryogenesis will be erased by sphaleron B + L violation solong as the φ condensate decays at T > T ew . In this case it is possible for the initial phaseof φ to take its natural value, θ ≈
1, without requiring a suppression of the flat directionbeyond that required to evade proton decay. Assuming that φ oscillations dominate the en-ergy density when the φ field decays to radiation, the energy density is given by ρ ≈ m φ φ ,where φ d is the amplitude of the oscillations when they decay. | φ d | is then related to thedecay temperature T d by | φ d | = k d T m φ ; k d = π g ( T d )30 . (53)For h | φ d | > m φ , where h is the gauge or Yukawa coupling of MSSM particles to the flatdirection, particles coupling to φ gain masses greater than m φ and so the φ decay is kinemat-ically suppressed. Therefore the condensate decays once φ ≈ m φ /h , assuming that Γ d > H ρ ≈ m φ /h . Therefore thedecay temperature, which is equivalent to the reheating temperature T R , is T d ≈ m φ ( h k d ) / . (54)With g ( T D ) ≈
200 we find k d ≈
65. Therefore T d ≈ . (cid:16) m φ (cid:17) (cid:18) . h (cid:19) / TeV , (55)where the particles with the smallest coupling h to φ will dominate the decay process, solong as Γ d > H . Therefore T d ≈ −
100 TeV in this model, assuming that the smallestcoupling satisfies 0 . > ∼ h > ∼ − . Once hφ < m φ the φ decay rate may be estimated to beΓ d ≈ h m φ / π , so the condition Γ d > H ≈ T /M is easily satisfied for T d in this range.We have assumed that the kinematic suppression of the decay rate prevents φ decayinguntil m φ > ∼ hφ , in which case T d < ∼
100 TeV. We should check that φ decay through heavyintermediate particles cannot cause it to decay significantly earlier. The decay rate via heavyintermediate particles of mass hφ will have the generic formΓ d ≈ α d m rφ ( hφ ) r , (56)where α d < r ≥ r = 4, and using equation (53), (56) andΓ d ≈ H ( T d ) ≈ T /M , this gives for the decay temperature T d ≈ (cid:18) α d g k k T (cid:19) / (cid:0) m φ M (cid:1) / ≈ (cid:18) α d g k k T (cid:19) / (cid:16) m φ (cid:17) / TeV . (57)Thus for typical couplings, the decay through intermediate states will also result in a re-heating temperature in the range T R ≈ −
100 TeV. It is significant that the reheat-ing temperature, T R < ∼
100 TeV, is naturally compatible with the thermal gravitino upperbound, T R < ∼ GeV, without any tuning of couplings. Even though the inflaton is part ofthe MSSM sector, it still leads to the required low reheating temperature. Since sphaleron B + L violation will erase the baryon asymmetry produced by the flat direction inflaton de-cay, baryogenesis must occur via some other mechanism, such as Affleck-Dine baryogenesisalong an orthogonal flat direction. 16 V. CONCLUSIONS
We have shown that it is possible for an MSSM extra-flat direction (one suppressed by aneffective mass scale much larger than the Planck mass) of the form
QQQL or u c u c d c e c to playthe role of the inflaton in a D -term inflation model. This eliminates the otherwise unmo-tivated singlet inflaton, reducing the number of required additional fields and so providinga minimal version of D -term inflation. The model has all the advantages of conventional D -term inflation with respect to compatibility with SUGRA and absence of fine-tuned cou-plings. The nature of reheating depends on whether the extra-flat direction is unstable withrespect to Q -ball formation. In the case where Q -balls form, it is possible to generate thebaryon asymmetry via Q -ball decay so long as the mass scale suppressing the flat directionis sufficiently large, Λ > ∼ − GeV, depending on the random phase θ of the flat direc-tion scalar during inflation. With Λ ≈ GeV and θ ≈ − it is possible to generate anobservably large baryon isocurvature perturbation. The reheating temperature from Q -balldecay is typically in the range 3 −
100 GeV. As this can be less than the neutralino LSPfreeze-out temperature, it is also possible to produce non-thermal dark matter from Q -balldecay. In the case where the flat direction has a large t squark component, there is no Q -ball formation. In this case the reheating temperature from decay of the homogeneous flatdirection condensate is in the range 1 −
100 TeV, ensuring sphaleron erasure of the baryonasymmetry from the B − L conserving directions while remaining naturally compatible withthe thermal gravitino upper bound on T R . The fact that we are able to calculate the re-heating temperature in this case is a direct consequence of the inflation being part of theMSSM sector. Since the baryon asymmetry from the flat direction is erased, the mass scalesuppressing the flat direction in this case is constrained only by proton decay, Λ > ∼ GeV.We have interpreted the WMAP observation of the spectral index as being due to an order10% CMB contribution from cosmic strings combined with a nearly scale-invariant adiabaticcurvature perturbation, n s ≈
1. As in conventional D -term inflation, we can simultaneouslysuppress the contribution of the cosmic strings to the required level while increasing n s byconsidering a small enough coupling of the inflaton to the Fayet-Iliopoulos charged fields.The extra-flat direction D -term inflation model has an advantage over conventional D -terminflation in that the range of coupling which leads to an order 10% contribution from cosmicstrings is much wider, making it perhaps more natural. For this solution to work, it is also17ecessary to have a U (1) gauge coupling that is somewhat smaller than the known gaugecouplings, g ≈ . − .
03. A significant feature of the model is that the superpotentialcoupling of the monomial
QQQL or u c u c d c e c to Φ + Φ − must be much larger than the puremonomial superpotential coupling. This feature may serve to test the compatibility of themodel with an ultra-violet complete theory, as we would naively expect all the superpotentialcouplings of the monomial to be strongly suppressed. Finally, we note that other solutions tothe cosmic string and spectral index problems are possible, for example SUGRA correctionsfrom a non-minimal K¨ahler potential [35] and/or modification of the inflaton potential byother fields, such as a RH sneutrino [36].The existence of extra-flat directions of the MSSM is one way to interpret the empiricalsuppression of non-renormalisable MSSM superpotential terms demanded by proton stabil-ity. It will be important to establish whether extra-flat directions can be understood inthe context of an ultra-violet complete theory and to explore more generally their role andpossible signatures in cosmology. Acknowledgments
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