Constraining Affleck-Dine Leptogenesis after Thermal Inflation
PPrepared for submission to JCAP
Constraining Affleck-Dine Leptogenesisafter Thermal Inflation
Seolhwa Kim and Ewan D. Stewart
Department of Physics, KAIST291 Daehak-ro, Yuseong-gu, Daejeon, KoreaE-mail: [email protected], [email protected]
Abstract.
Affleck-Dine leptogenesis after thermal inflation along the LH u direction requires m L + m H u < | L | (cid:39) | H u | ∼ GeV). We renormalised this condition fromthe AD scale to the soft supersymmetry breaking scale by solving the renormalisation groupequations perturbatively in the Yukawa couplings to obtain a semi-analytic constraint on the softsupersymmetry breaking parameters. We also used a fully numerical method to renormalise thebaryogenesis condition and constrained the Minimal Supersymmetric Cosmological Model usingthe resulting baryogenesis condition and other constraints, specifically, electroweak symmetrybreaking, the observed Higgs mass, and the axino dark matter abundance. a r X i v : . [ h e p - ph ] M a y ontents H u H d constraint 15 B.1 Zeroth order solution 21B.2 First order solution 22B.3 Numerical coefficients of the baryogenesis condition 24
Cosmological moduli fields [1–3] are scalar fields with Planckian vacuum expectation values.They have a vanishing potential when supersymmetry is unbroken and develop a potential onlyafter supersymmetry breaking. In the early Universe, the moduli fields acquire a potential V cos (Φ) = ρ f (cid:18) Φ M Pl (cid:19) = α H (Φ − Φ ) + · · · (1.1)where H is the Hubble parameter and α ∼ O (1). When H ∼ m Φ , their vacuum potential V vac (Φ) = m M g (cid:18) Φ M Pl (cid:19) = 12 m (Φ − Φ ) + · · · (1.2)– 1 –ecomes significant. The minimum is then shifted toward Φ , and the moduli fields undergohomogeneous oscillations around Φ with an amplitude Φ ∼ Φ − Φ ∼ M Pl . This coherentoscillation may get critically damped initially as H (cid:46) m Φ , and its large relic density n Φ s ∼ Φ g / ∗ m / M / ∼ (cid:18) g ∗ (cid:19) (cid:18) TeV m Φ (cid:19) (cid:18) Φ M Pl (cid:19) (1.3)where g ∗ is a relativistic degrees of freedom, leads to a matter domination until its late timedecay given by the lifetime τ = Γ − = (cid:18) α Φ m M (cid:19) − ∼ s α − (cid:18) TeV m Φ (cid:19) (1.4)and characterized by the low decay temperature T ∼ g − ∗ Γ M Pl = α / m / g / ∗ M / ∼
10 keV α Φ (cid:18) g ∗ (cid:19) (cid:16) m Φ TeV (cid:17) (1.5)This late time decay into energetic photons or hadronic showers around or after Big Bangnucleosynthesis (BBN) can destroy previously formed nuclei. In order not to upset BBN, thedecay temperature must satisfy T (cid:38) n Φ s (cid:46) − (cid:18) TeV m Φ (cid:19) (1.6)There have been some attempts to address this problem. First, if the moduli mass islarger than 10 to 100 TeV, it decays earlier than BBN and hence does not affect BBN [7, 8].Second, Ref. [9] suggested diluting the moduli abundance through a rolling inflation. However,the scale is typically too high so that moduli are regenerated after the inflation. Lastly, thermalinflation [10–17] provides enough inflation to dilute moduli at a scale sufficiently low to avoidtheir regeneration. The low energy effective potential of the flaton is V = V − m φ | φ | + · · · (1.7)where the vacuum expectation value of the flaton satisfies m φ (cid:28) φ (cid:28) M Pl , and V ∼ m φ φ (cid:28) m φ M so that there is no slow roll inflation. Thermal inflation starts when the flaton is heldat the origin and the thermal energy density ( ∼ T ) falls below V . It ends at the criticaltemperature when the flaton rolls away from the origin ( T ∼ m φ ). Thus, thermal inflationoccurs when the temperature satisfies m φ (cid:46) T (cid:46) V (1.8)The number of e-folds of thermal inflation is therefore N = ln a f a i = ln T i T f (cid:39) ln V / m φ (cid:39)
12 ln φ m φ ∼
10 (1.9)– 2 –or φ ∼ GeV and m φ ∼ a f and a i are scale factors at the beginning and endof thermal inflation respectively. This moderately redshifts the density perturbation from theprimordial inflation.After thermal inflation, the flaton decays with Γ = βm φ /φ . Assuming decay productsthermalize promptly, the decay temperature is T d ∼ g − ∗ Γ M Pl (cid:39) β (cid:18) g ∗ (cid:19) − (cid:18) GeV φ (cid:19) (cid:16) m φ TeV (cid:17) (1.10)For the decay to precede BBN, one requires T d (cid:38) φ (cid:46) GeV for m φ ∼ s d R d s c R c ∼ ρ φd R d /T d g ∗ c R c T c ∼ V g ∗ c T c T d ∼ (cid:18) g ∗ c (cid:19) (cid:32) V / GeV (cid:33) (cid:18) TeV T c (cid:19) (cid:18) GeV T d (cid:19) (1.11)where the subscript b , c and d denote at the beginning of and the end of thermal inflation,and at the decay of the flaton respectively. Combined with Eq. (1.3), the moduli abundance isdiluted to n Φ s (cid:12)(cid:12)(cid:12) d = 1∆ n Φ s (cid:12)(cid:12)(cid:12) a ∼ − (cid:16) g ∗ (cid:17) (cid:18) TeV m Φ (cid:19) (cid:18) Φ M Pl (cid:19) (cid:32) GeV V / (cid:33) (cid:18) T c TeV (cid:19) (cid:18) T d GeV (cid:19) . (1.12)where the subscript a denotes when H ∼ m Φ . Since this is greater than the bound in Eq. (1.6),a sufficient dilution may require double thermal inflation [11, 18–20].Meanwhile, the thermal inflationary potential energy is moduli field-dependent. This ap-pears in the potential of the moduli fields as V = 12 m (Φ − Φ ) − α (cid:48) V M Pl Φ + · · · (1.13)Hence, the moduli fields begin to oscillate after thermal inflation with an amplitude δ Φ ∼ α (cid:48) V m M Pl (1.14)However, the corresponding moduli abundance n Φ s ∼ n Φ c s d n Φ d n Φ c ∼ m Φ δ Φ s d ρ d V ∼ α (cid:48) V T d m M (1.15)= 10 − α (cid:48) (cid:32) V / GeV (cid:33) (cid:18) T d GeV (cid:19) (cid:18)
TeV m Φ (cid:19) (1.16)is safe. – 3 – .3 Affleck-Dine leptogenesis after thermal inflation The low scale of thermal inflation V / ∼ GeV and the low reheat temperature T d ∼ M GUT ∼ GeV and m ¯ ν (cid:38) GeV [21], which decaybefore thermal inflation. Hence, any baryon asymmetries from their decays would be diluted bythermal inflation to a negligible amount. Likewise, the standard Affleck-Dine (AD) mechanism[22, 23] in gravity-mediated supersymmetry breaking scenarios occurs before thermal inflationwhen the Hubble parameter is comparable to the sparticle masses. These difficulties suggestthat baryogenesis should occur during the thermal inflationary era and early works in thisdirection are given in Ref. [25–27].This paper focuses on AD leptogenesis after thermal inflation [18, 19, 28–32] implementedin the Minimal Supersymmetric Cosmological Model (MSCM) [19]. The MSCM is a minimalimplementation of supersymmetry, thermal inflation and baryogenesis with the QCD axion[33, 34] and axino [35, 36] as dark matter. The superpotential of the MSCM is W = λ u QH u ¯ u + λ d QH d ¯ d + λ e LH d ¯ e + 12 κ ν ( LH u ) + κ µ φ H u H d + λ χ φχ ¯ χ (1.17)where κ ν ( LH u ) provides the neutrino mass m ν = | κ ν H u | = | κ ν | v sin β (1.18)and µ = κ µ φ is the MSSM µ -parameter. The coupling λ χ renormalises the flaton’s mass tobecome negative at the origin and couples the flaton to the thermal bath, holding it at the originduring thermal inflation. As well as being the flaton whose potential drives thermal inflation, φ also acts as the Peccei-Quinn field containing the QCD axion.In the MSCM, we obtain a lepton asymmetry by the AD mechanism along the LH u direction. For the LH u direction to have an initially large field value, it is necessary to have m LH u ≡ (cid:0) m L + m H u (cid:1) < | L | ∼ | H u | (cid:46) AD scale for some lepton generation. Eq. (1.19) is most easily satisfied for thethird generation and so we take L = L in this paper. Moreover, since m L H u becomes morenegative at lower scales, it is sufficient to require m L H u (cid:12)(cid:12) AD < L = (cid:18) l (cid:19) , H u = (cid:18) h u (cid:19) , H d = (cid:18) h d (cid:19) , φ = φ (1.21)and other fields zero, the MSCM potential reduces to V = V + ˜ m φ | φ | + m L | l | + m H u | h u | + m H d | h d | + (cid:18) A ν κ ν l h u − Bκ µ φ h u h d + c.c. (cid:19) + (cid:12)(cid:12) κ ν lh u (cid:12)(cid:12) + (cid:12)(cid:12) κ ν l h u − κ µ φ h d (cid:12)(cid:12) + (cid:12)(cid:12) κ µ φ h u (cid:12)(cid:12) + | κ µ φh u h d | + g (cid:0) | h u | − | h d | − | l | (cid:1) (1.22) We refer readers to Ref. [24] for the AD baryogenesis assuming gauge mediated supersymmetry breaking. – 4 –here ˜ m φ ( φ ) is the renormalised mass of the flaton which runs from ˜ m φ > m φ < LH u flat direction becomes unstable due to Eq. (1.20). Thecorresponding minimum A ν κ ν l h u = −| A ν κ ν l h u | (1.23) | l | (cid:39) | h u | (cid:39) (cid:118)(cid:117)(cid:117)(cid:116) | A ν | + (cid:113) | A ν | − m LH u | κ ν | (cid:39) GeV (cid:115)(cid:18) | m LH u | TeV (cid:19) (cid:18) − eV m ν (cid:19) (1.24)provides the initial condition for Affleck-Dine leptogenesis.After thermal inflation ends, as the flaton begins to roll away from the origin, two termsin the potential, Bκ µ φ h u h d and ( κ ν l h u ) ∗ κ µ φ h d + c.c., give rise to a non-zero field value of h d . When φ approaches to φ , the term (cid:12)(cid:12) κ µ φ h u (cid:12)(cid:12) gives a positive contribution to the masssquared in the LH u direction at the origin, pulling l, h u and h d towards the origin. Meanwhile,the terms Bκ µ φ h u h d and ( κ ν l h u ) ∗ κ µ φ h d + c.c. tilt the potential in the phase direction. Thischanges the phase of lh u and hence produces a lepton asymmetry.The amplitude of the homogeneous mode is damped due to preheating and friction inducedby the thermal bath. Therefore, the lepton number violating terms become less significant sothat the lepton number is conserved. After the AD field’s preheating and decay, the associatedpartial reheat temperature allows sphaleron processes that convert the lepton number to abaryon number. In Section 2, we will solve the renormalisation group equations to translate the baryogenesiscondition from the AD scale to the soft supersymmetry breaking scale and obtain semi-analyticconstraints on the soft supersymmetry breaking parameters. Then, we will compare the semi-analytic formula to results using the numerical package FlexibleSUSY [37]. In Section 3, wewill assume CMSSM boundary conditions and combine the baryogenesis constraint with otherconstraints–electroweak symmetry breaking, the Higgs mass, the axino cold dark matter abun-dance and the stability of the H u H d direction.In Appendix A, we will address the connection between a field value and the renormalisa-tion scale through the renormalisation group improvement. In Appendix B, we give the detailedcalculation for the semi-analytic formula of the baryogenesis condition. The baryogenesis condition m L H u (cid:12)(cid:12) AD < | L | ∼ | H u | ∼ GeV. In Appendix A, we will explain howthe renormalisation group improvement connects the renormalisation of couplings in field spacewith the renormalisation with respect to the renormalisation scale. Resorting to this, we solvethe renormalisation group equations with respect to the renormalisation scale and obtain thebaryogenesis condition imposed at the AD field value expressed in terms of the parameters atthe soft supersymmetry breaking scale, or alternatively, in terms of the universal CMSSM GUTparameters. – 5 –he relevant renormalisation group equations are x = 18 π log µm s (2.1) ddx g = −
335 (2.2) ddx g = − ddx g = 3 (2.4) M i ∝ g i , i = 1 , , ddx λ t = (cid:18) g + 3 g + 1315 g − λ b − λ ν (cid:19) λ t − ddx λ b = (cid:18) g + 3 g + 715 g − λ t − λ τ (cid:19) λ b − ddx λ τ = (cid:18) g + 95 g − λ b − λ ν (cid:19) λ τ − ddx A t = 6 λ t A t + λ b A b + λ ν A ν + 163 g M + 3 g M + 1315 g M (2.9) ddx A b = λ t A t + 6 λ b A b + λ τ A τ + 163 g M + 3 g M + 715 g M (2.10) ddx A τ = 3 λ b A b + 4 λ τ A τ + λ ν A ν + 3 g M + 95 g M (2.11) ddx D t = 6 λ t A t + λ b A b + λ ν A ν − g M − g M − g M + 6 λ t D t + λ b D b + λ ν D ν (2.12) ddx D b = λ t A t + 6 λ b A b + λ τ A τ − g M − g M − g M + λ t D t + 6 λ b D b + λ τ D τ (2.13) ddx D τ = 3 λ b A b + 4 λ τ A τ + λ ν A ν − g M − g M + 3 λ b D b + 4 λ τ D τ + λ ν D ν (2.14) ddx (cid:0) m L + m H u (cid:1) = 3 λ t A t + λ τ A τ ++2 λ ν A ν − g M − g M +3 λ t D t + λ τ D τ +2 λ ν D ν (2.15)where D t ≡ m H u + m Q + m t , D b ≡ m H d + m Q + m b , (2.16) D τ ≡ m H d + m L + m τ , D ν ≡ m H u + m L + m ν (2.17)Purple terms are from the right-handed neutrino which we neglect until Section 3.3,– 6 –hese equations take the form of ddx y = f ( x ) y + g ( x ) (2.18)which has the solution y ( x ) = e (cid:82) x f ( t ) dt (cid:90) x g ( t ) e − (cid:82) t f ( t (cid:48) ) dt (cid:48) dt (2.19)Using this, the gauge couplings and the gaugino masses can be solved exactly g = g (0)1 − g (0) x , g = g (0)1 − g (0) x , g = g (0)1 + 3 g (0) x (2.20) M = M (0)1 − g (0) x , M = M (0)1 − g (0) x , M = M (0)1 + 3 g (0) x (2.21)Noting the hierarchy in Yukawa couplings λ t (cid:39) . × (cid:18) β (cid:19) , λ b (cid:39) . × − (1 + tan β ) , λ τ (cid:39) . × − (1 + tan β )(2.22)i.e. λ τ < λ b < λ t ∼ . β <
42, we will use perturbation in λ b and λ τ to solve theremaining renormalisation group equations for m L H u . λ (0) t → λ b → λ τ , λ (1) t ↓ ↓ ↓ A (0) t → A b → A τ , A (1) t ↓ ↓ ↓ D (0) t → D b → D τ , D (1) t ↓ ↓ (cid:0) m L + m H u (cid:1) (0) (cid:0) m L + m H u (cid:1) (1) Table 1 : Sequence of solving for m L H u perturbatively in λ b , λ τ In the zeroth order, we set λ b = λ τ = 0. This amounts to neglecting the colored terms inEqs. (2.6), (2.9), (2.12) and (2.15). First, we solve Eq. (2.6) for λ (0) t . Then, we substitute theresulting λ (0) t into Eq. (2.9) and solve for A (0) t . Next, we substitute the resulting A (0) t and λ (0) t into Eq. (2.12) and solve for D (0) t . Finally, we substitute λ (0) t , A (0) t and D (0) t into Eq. (2.15) andsolve for m L H u . The sequence above can be summarized as λ (0) t → A (0) t → D (0) t → m LH u (firstcolumn of Table 1). We put the detailed calculations in Appendix B.1. The resulting zerothorder analytic expression is (cid:0) m L + m H u (cid:1) (0) (cid:12)(cid:12)(cid:12) AD = m L (0) + m H u (0) + (cid:88) X α (0) X X (0) , with X ∈ { D t , A t , A t M i , M i M j | i = 1 , , } (2.23)– 7 –here the coefficients α (0) X are functions of g i (0) , λ t (0) and x and given in Tables 5 to 9 inAppendix B.3. For example, the baryogenesis condition for tan β = 20 and the AD scale at 10 GeV ( x = 0 . > (cid:0) m L + m H u (cid:1) (0) (cid:12)(cid:12)(cid:12) AD = m L + m H u + 0 . A t + A t (0 . M + 0 . M + 0 . M ) + 0 . D t − . M − . M − . M + 0 . M M (2.24)where all parameters on the right-hand side are evaluated at the soft supersymmetry breakingscale.Since the M contribution is small compared to other numerical coefficients in Table 5 to9, we set M = 0 to give the simplified zeroth order semi-analytic expression (cid:0) m L + m H u (cid:1) (0)S (cid:12)(cid:12)(cid:12) AD = m L (0) + m H u (0) + (cid:88) X α (0) X X (0) , with X ∈ { D t , A t , A t M i , M i M j | i = 2 , } (2.25)For example, for tan β = 20 and the AD scale at 10 GeV, the simplified zeroth order baryoge-nesis condition is0 > (cid:0) m H u + m L (cid:1) (0)S (cid:12)(cid:12)(cid:12) AD = m L + m H u + 0 . A t + A t (0 . M + 0 . M ) + 0 . D t − . M − . M + 0 . M M (2.26)Alternatively, one can replace x = 0 in the integral domain of Eq. (B.10) with the GUTscale to express the zeroth order semi-analytic formula of the baryogenesis condition in termsof the universal CMSSM GUT scale parameters as0 > (cid:0) m L + m H u (cid:1) (0) (cid:12)(cid:12)(cid:12) AD = 1 . m − . A + 0 . A M / + 0 . M / (2.27) At first order, we follow the second and third columns in Table 1. First, we set λ τ = 0 (neglectingorange terms) and solve Eqs. (2.7), (2.10) and (2.13) for λ b → A b → D b using the zeroth ordervalues of λ (0) t , A (0) t and D (0) t in the same way we solved for λ (0) t → A (0) t → D (0) t at zeroth order.Next, we substitute λ b and λ (0) t into Eq. (2.8) and solve for λ τ . Then, we follow the samesequence in the previous step to solve Eqs. (2.8), (2.11) and (2.14) for λ τ → A τ → D τ (green inthe third column of Table 1). In the same way, we solve Eqs. (2.6), (2.9) and (2.12) for λ (1) t , A (1) t and D (1) t with λ b , A b , and D b calculated before (blue in the third column of Table 1). Finally,we combine D b , D τ and D (1) t to compute m L H u .We put the detailed calculations in Appendix B.2 and the resulting zeroth and first orderexpressions of m L + m H u in Tables 5 to 9. For example, for tan β = 20 and the AD scale at10 GeV ( x = 0 . > (cid:0) m L + m H u (cid:1) (1) (cid:12)(cid:12)(cid:12) AD = m L + m H u + 0 . A t + A t (0 . M + 0 . M + 0 . M ) + 0 . D t − . M − . M − . M + 0 . M M + 0 . M M + 0 . M M + 0 . A b M + 0 . A τ + 0 . D τ From Tables 5 to 9, the coefficients of the zeroth and the first order semi-analytic formulae areclose, which gives confidence to this perturbative calculation.– 8 – L + m H u (AD = 10 GeV) [ TeV] tan β A [TeV] FlexibleSUSY First order Zeroth order5 0 16 .
93 15 .
69 15 .
575 16 .
67 15 .
29 15 . .
62 9 .
90 9 . .
34 16 .
12 15 .
725 17 .
07 16 .
09 15 . .
08 10 .
44 9 . .
27 16 .
04 15 .
875 16 .
94 16 .
02 15 . .
72 10 .
30 8 . .
96 15 .
75 14 .
795 16 .
50 15 .
70 14 . .
81 10 .
83 7 . .
40 15 .
30 13 .
035 15 .
74 15 .
10 13 . .
31 8 .
73 5 . Table 2 : m L + m H u (AD = 10 GeV) from FlexibleSUSY, the zeroth and first order semi-analytic formulae for m = 3 TeV , M / = 4 TeV and µ > To compare with the numerical package FlexibleSUSY, we reduce the parameter space tothat of the constrained MSSM (CMSSM) which assumes a universal scalar mass, m , gaug-ino mass, M / , and trilinear coupling, A at the GUT scale, with B and | µ | replaced bythe electroweak symmetry breaking scale and tan β . Table 2 shows the numerical values of( m L + m H u )(10 GeV) from the following three methods(1) the numerical package FlexibleSUSY(2) the first order semi-analytic formulae, Eq. (B.24)(3) the zeroth order semi-analytic formulae, Eq. (2.23)assuming CMSSM boundary conditions with m = 3 TeV , M / = 4 TeV , A = 0 , ,
10 TeVand tan β = 5 , , , ,
40. For the two semi-analytic formulae, we took the low scale MSSMparameters ( D t (0) , A t (0) , · · · . ) from FlexibleSUSY. Except for tan β = 5 and A = 5 TeV, thefirst order result is closer to the numerical result than the zeroth order result. Also, the differencebetween the first and zeroth order results increases as tan β increases as can be expected forthe perturbative method.In Figure 1, we plotted various versions of the baryogenesis condition on the CMSSM pa-rameter space. Cyan corresponds to m L H u > igure 1 : The CMSSM parameter space constrained by the baryogenesis condition and elec-troweak symmetry breaking. From top to bottom, tan β = 10 , ,
30. From left to right, m = 0 , , , (cid:4) m L H u (cid:12)(cid:12) m s > (cid:4) m L H u (cid:12)(cid:12) AD > (cid:4) m L H u S (cid:12)(cid:12)(cid:12) AD > (cid:4) m L H u (cid:12)(cid:12)(cid:12) AD > (cid:4) electroweak symmetry isunbroken or incorrectly broken.out by the simplified zeroth order semi-analytic renormalised baryogenesis condition. Green,yellow, and cyan regions are ruled out by the numerically calculated renormalised baryogenesiscondition. Purple, green, yellow and cyan regions are ruled out by the zeroth order baryoge-nesis condition given in Eq. (2.27). From Eq. (2.27), it is understandable that the slope ofthe boundary of the baryogenesis condition is straight for m = 0 and curved for m (cid:54) = 0.Black regions are ruled out by electroweak symmetry breaking. One cans see that electroweaksymmetry breaking and the baryogenesis constraints are complementary. Meanwhile, the tree-level result is much weaker than the renormalised constraints, showing the importance of therenormalisation. Also, the renormalised constraints are similar, showing the robustness of thesemi-analytic formulae. – 10 – Constraining the MSCM
In the MSCM of Eq. (1.17), the cold dark matter consists of axions [33, 34] and axinos [35, 36].The decay temperature of the flaton after thermal inflation is (Eq. (94) in Ref. [19]) T d (cid:39)
100 GeV (cid:12)(cid:12)(cid:12)(cid:12) m A − | B | m A (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) GeV φ (cid:19) (cid:18) | µ | TeV (cid:19) (cid:18) GeV m PQ (cid:19) (cid:34) f (cid:32) m h m (cid:33)(cid:35) (3.1)= 10 GeV C (cid:18) GeV φ (cid:19) (cid:18) | µ | TeV (cid:19) (3.2)where C can be estimated as O (1) from Figure 6 in Ref. [19]. Axion abundance
For T d (cid:29) a (cid:39) . (cid:18) f a GeV (cid:19) . (3.3) φ can be expressed in terms of f a , φ = N √ f a (cid:39) × GeV (cid:18) N (cid:19) (cid:18) f a GeV (cid:19) (3.4)where N = (cid:80) i p i and p i is the PQ charge of the i th quark. Axino from flaton decay
The effective interaction between the axino and the radial flatonis [19] α ˜ a m ˜ a √ φ δr ˜ a + c.c. (3.5)where m ˜ a is the mass of the axino. Using the flaton to axinos decay rateΓ φ → ˜ a ˜ a = α a m a m PQ πφ (3.6)one can estimate the current abundance of axinos produced by the flaton decay as (Eq. (142)in Ref. [19])Ω φ → ˜ a (cid:39) .
023 Γ / φ Γ / (cid:18) g ∗ ( T d ) (cid:19) (cid:16) α ˜ a − (cid:17) (cid:16) m ˜ a GeV (cid:17) (cid:18)
10 GeV T d (cid:19) (cid:18) × GeV φ (cid:19) (3.7)where Γ SM is the rate of the flaton’s decay to the Standard Model particles, and Γ φ (cid:39) Γ SM isthe total decay rate of the flaton. Axino from NLSP decay
The NLSP decays to axinos with the decay rateΓ N → ˜ a = A m N πφ (3.8)– 11 –here A ∼ O (1) and m N is the mass of the NLSP, producing the axino abundance (Eq. (166)in Ref. [19])Ω N → ˜ a ∼ . A Γ / Γ / φ (cid:32) g ∗ ( T d ) / g ∗ ( T N ) / (cid:33) (cid:16) m N GeV (cid:17) (cid:16) m ˜ a GeV (cid:17) (cid:18) GeV φ (cid:19) (cid:18) T d m N (cid:19) (3.9)= 0 . C (cid:16) m N TeV (cid:17) (cid:18) × GeV φ (cid:19) (cid:18) T d m N (cid:19) (3.10)where C ∼ O (1).Assuming the axino abundance is less than the axion abundance0 . C (cid:16) m N TeV (cid:17) (cid:18) × GeV φ (cid:19) (cid:18) T d m N (cid:19) (cid:46) Ω ˜ a (cid:46) . m N (cid:38)
40 GeV C C (cid:18) | µ | TeV (cid:19) (cid:18) × GeV φ (cid:19) (3.12)Referring to Eqs. (3.3) and (3.4), we will use f a (cid:46) GeV and hence φ (cid:46) × GeV inthe following discussion. Then, the mass of the NLSP reduces to m N (cid:38)
40 GeV (cid:18) | µ | TeV (cid:19) (3.13) In Figure 2 and 3, we imposed the following constraints on the CMSSM parameters usingFlexibleSUSY(1) Electroweak symmetry breaking(2) Baryogenesis condition m L H u (cid:12)(cid:12) AD < .
68 GeV < m h < .
68 GeV (3.15)(4) Axino dark matter abundance m N (cid:38)
40 GeV (cid:18) | µ | TeV (cid:19) (3.13)The dark matter constraint allows parameters inside the white, yellow, cyan and green(i.e. colors excluding magenta) oval in the middle. For tan β ≥
10, the Higgs mass constraint issatisfied inside the white, yellow, magenta and red (i.e. colors excluding cyan) tick ( (cid:88) ) extendingfrom the bottom left to the top right. The baryogensis condition is satisfied in the white, cyan,magenta and purple (i.e. colors excluding yellow) bands next to the regions prohibited bythe electroweak symmetry breaking constraint. All four constraints are satisfied in the whiteoverlapping regions. – 12 – igure 2 : The CMSSM parameter space with four constraints. From top to bottom, tan β =5 , , , , ,
40. From left to right, m = 0 , , (cid:4) electroweak symmetry is incorrectlybroken, (cid:4) m L H u (cid:12)(cid:12) AD > (cid:4) m h < .
68 GeV or m h > .
68 GeV (i.e. the Higgs mass constraint is violated), (cid:4) m N (cid:46)
40 GeV ( µ/ TeV) (i.e.the axino dark matter constraint is violated). (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) + (cid:4) .White is the only allowed region satisfying all constraints.– 13 – igure 3 : CMSSM parameter space with four constraints. From top to bottom, tan β =5 , , , , ,
40. From left to right, m = 3 , , (cid:4) electroweak symmetry is incorrectlybroken, (cid:4) m L H u (cid:12)(cid:12) AD > (cid:4) m h < .
68 GeV or m h > .
68 GeV (i.e. the Higgs mass constraint is violated), (cid:4) m N (cid:46)
40 GeV ( µ/ TeV) (i.e.the axino dark matter constraint is violated). (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) + (cid:4) .White is the only allowed region satisfying all constraints.– 14 – .3 H u H d constraint Since m L >
0, the baryogenesis condition m L H u (cid:12)(cid:12) AD < m H u <
0. However, this may cause H u H d to become large instead of LH u . To avoid this, we require m H u H d (cid:12)(cid:12) AD > ddx m L = λ τ A τ + λ ν A ν − g M − g M − g S + λ τ D τ + λ ν D ν (3.17) ddx m H u = 3 λ t A t + λ ν A ν − g M − g M + 310 g S + 3 λ t D t + λ ν D ν (3.18) ddx ( m H d − m L ) = 3 λ b ( A b + D b ) − λ ν ( A ν + D ν ) (3.19)where purple terms are from right-handed neutrinos. Eq. (3.19) implies that m H u H d < m L H u in the CMSSM without right-handed neutrinos. Hence, m L H u < m H u H d > m L H u < m H u H d > H u H d direction, one should relax the CMSSM boundary conditions or include right-handed neutrinosto the theory.We adopt the ν CMSSM [38] to consider the effect of right-handed neutrinos from M GUT to M ¯ ν and integrated them out below M ¯ ν . Because M ¯ ν is close to M GUT , we simply integratedtheir effects linearly in Eqs. (3.17) and (3.18). Namely, we modified the CMSSM boundaryconditions for m L and m H u as m L , m H u = m → m + λ ν π (cid:0) A + 3 m (cid:1) log M ¯ ν M GUT (3.20)and generated the low scale parameters using FlexibleSUSY. We also assumed M ¯ ν = 10 GeVand λ ν = λ t at M GUT = 1 . × GeV. Meanwhile, we neglected the effects of right-handedneutrinos on the Yukawa and trilinear couplings.Extending the baryogenesis condition to the combination of Eqs. (1.20) and (3.16), weused the same four constraints—the baryogenesis condition, electroweak symmetry breaking,Higgs mass and the axino dark matter abundance—to constrain the CMSSM parameter space.Figures 4 and 5 show the resulting CMSSM parameter space.From Figures 4 and 5, m H u H d (cid:12)(cid:12)(cid:12) AD > m L H u (cid:12)(cid:12) AD < m L H u more negative atlow scales including AD scale, relaxing the previous baryogenesis condition ( m L H u (cid:12)(cid:12) AD < igure 4 : The CMSSM parameter space with four constraints in the presence of the right-handed neutrino. From top to bottom, tan β = 5 , , , ,
30. From left to right, m =0 , , (cid:4) electroweak symmetry is incorrectly broken, (cid:4) m L H u (cid:12)(cid:12) AD > m H u H d (cid:12)(cid:12)(cid:12) AD < (cid:4) m h < .
68 GeV or m h > .
68 GeV (i.e. Higgsmass constraint is violated), (cid:4) m N (cid:46)
40 GeV ( µ/ TeV) (i.e. the axino dark matter constraintis violated). (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) + (cid:4) . White is the only allowed regionsatisfying all constraints. A spectrum for each central value is given at Figure 6 and Tables 3and 4. – 16 – igure 5 : The CMSSM parameter space with four constraints in the presence of the right-handed neutrino. From top to bottom, tan β = 5 , , , ,
30. From left to right, m =3 , , (cid:4) electroweak symmetry is incorrectly broken, (cid:4) m L H u (cid:12)(cid:12) AD > m H u Hd (cid:12)(cid:12) AD < (cid:4) m h < .
68 GeV or m h > .
68 GeV (i.e. Higgsmass constraint is violated), (cid:4) m N (cid:46)
40 GeV ( µ/ TeV) (i.e. the axino dark matter constraintis violated). (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) , (cid:4) = (cid:4) + (cid:4) + (cid:4) . White is the only allowed regionsatisfying all constraints. – 17 – a) Left-hand side region (b) Right-hand side region Figure 6 : Sparticle spectra for the central points on the left- and right-hand side regionssatisfying all four constraints in Figure 4. Higgs scalar mass eigenstates consist of h , H : CPeven neutral scalars, A : CP odd neutral scalar, H ± : CP odd charge +1 scalars. ˜ g : gluino, N i : neutralinos, C j : charginos, ˜ u k : up-type squarks, ˜ d k : down-type squarks, ˜ e k : sleptons, ˜ ν l :sneutrinos where i = 1 , , , j = 1 , k = 1 , · · · , l = 1 , ,
3, and each index is in ascendingorder of pole mass. h H A H ± ˜ u ˜ u ˜ u ˜ u ˜ u ˜ u N N N N ˜ d ˜ d ˜ d ˜ d ˜ d ˜ d C C ˜ g ˜ e ˜ e ˜ e ˜ e ˜ e ˜ e ν ˜ ν ˜ ν Table 3 : TeV scale pole masses of sparticles for the central point on the left-hand sideregion satisfying all four constraint which corresponds to tan β = 10 , m = 1 TeV , M / =2 . , A = − . h H A H ± ˜ u ˜ u ˜ u ˜ u ˜ u ˜ u N N N N ˜ d ˜ d ˜ d ˜ d ˜ d ˜ d C C ˜ g ˜ e ˜ e ˜ e ˜ e ˜ e ˜ e ν ˜ ν ˜ ν Table 4 : TeV scale pole masses of sparticles for the central point on the right-hand sideregion satisfying all four constraint which corresponds to tan β = 15 , m = 2 TeV , M / =6 .
55 TeV , A = 14 .
45 TeV. – 18 –
Conclusion
Affleck-Dine leptogenesis after thermal inflation along the LH u direction [18, 19, 28–30] requires m LH u < | L | (cid:39) | H u | ∼ GeV). In Section 2, we renormalised thisbaryogenesis condition from the AD scale to the soft supersymmetry breaking scale by solvingthe renormalisation group equations perturbatively in the Yukawa couplings. The resultingzeroth and first order semi-analytic constraints on the soft supersymmetry breaking parametersare Eqs. (2.23) and (B.24) with the numerical coefficients given in Tables 5–9. Since the M contributions are small, we set M = 0 to get the simplified zeroth order formula given inEq. (2.25). The robustness of our formula can be seen in several ways. Firstly, the numericalcoefficients in the zeroth and first order formulae are close. Also, in Table 2, the numericalvalues of m L H u at the AD scale from the semi-analytic formulae and the numerical packageFlexibleSUSY are fairly close. Lastly, in Figure 1, the simplified zeroth order formula constrainsthe CMSSM parameter space similarly to the fully numerical result but much stronger than thetree-level formula.In Section 3, we used the numerical package FlexibleSUSY to renormalise the baryogene-sis condition assuming CMSSM boundary conditions. We considered the MSCM of Eq. (1.17)and combined the renormalised baryogenesis condition with other constraints, specifically, elec-troweak symmetry breaking, Higgs mass, Eq. (3.15), and the cold dark matter abundance,Eq. (3.13). In Figures 2 and 3, there is a region satisfying all the four constraints, which corre-sponds to10 (cid:46) tan β (cid:46) , (cid:46) m (cid:46) , (cid:46) M / (cid:46) , − . (cid:46) A (cid:46) − H u H d direction by adding the additionalconstraint m H u H d (cid:12)(cid:12)(cid:12) AD >
0, which is incompatible with m L H u (cid:12)(cid:12) AD < m H u H d (cid:12)(cid:12)(cid:12) AD > m L H u (cid:12)(cid:12) AD < (cid:46) tan β (cid:46) , (cid:46) m (cid:46) , . (cid:46) M / (cid:46) , − (cid:46) A (cid:46) − . (cid:46) tan β (cid:46) , (cid:46) m (cid:46) , (cid:46) M / (cid:46) . ,
10 TeV (cid:46) A (cid:46)
18 TeV(4.3)Lastly, we suggest to add an extra constraint—tunneling to non-MSSM vacua—for thefuture work. The baryogenesis condition, m L H u < L, H u , Q, ¯ d ) or ( L, H u , ¯ e ) [39, 40] to which the MSSM vacuum can tunnel to.Requiring the life time of the MSSM vacuum to be larger than the age of the Universe wouldconstrain the CMSSM parameters in a manner complementary to the baryogenesis conditionand similar to but stronger than the electroweak symmetry breaking condition. Acknowledgements
EDS thanks Gabriela Barenboim, Wan-il Park and Javier Rasero for collaboration on an earlierversion of this project and Kenji Kadota for helpful discussions. SK thanks Jae-hyeon Park– 19 –or helping with FlexibleSUSY. We thank Wan-il Park for helpful comments on a draft of thispaper.
Appendices
A Field-dependent renormalisation using RG improvement
The renormalization group improved potential satisfies [41] µ ddµ V ( µ, λ i , φ ) = (cid:18) µ ∂∂µ + β λ i ∂∂λ i − γφ ∂∂φ (cid:19) V ( µ, λ i , φ ) = 0 (A.1)where β λ i are beta functions of the couplings λ i and γ is the wavefunction renormalization.One can solve the Eq. (A.1) using the method of characteristic [42]. This method regards eachvariable µ, λ and φ as points on a curve parametrized by t , V ( µ, λ, φ ) = V ( µ ( t ) , λ ( t ) , φ ( t )) (A.2)where the variables satisfy dµdt = µ, dλ i dt = β λ i , dφdt = − γφ (A.3)with the solution µ ( t ) = µ e t , φ ( t ) = φ (0) exp (cid:18) − (cid:90) t γ ( λ i ( t (cid:48) )) dt (cid:48) (cid:19) (A.4)Then, the renormalization scale, field values and couplings are functions of t such that anyof their changes with respect to the t cancel each other so that the potential is left invariant.Hence, if the inverse map of Eq. (A.4) exists, one can relate the field value to the renormalizationscale. Any choice of t allows one to connect the renormalization scale to the field value. However,there is some choice of t that simplifies the RG improved potential in 1-loop order. For example,consider a 1-loop effective potential V − loop = 164 π STr (cid:18) M ( λ i ( t ) , φ ( t )) ln M ( λ i ( t ) , φ ( t )) µ e t − (cid:19) (A.5)with the dominant eigenvalues of M are close to each other. Let ¯ M denotes one of thedominant eigenvalues of M and choose tt = 12 ln ¯ M ( λ i ( t ) , φ ( t )) µ (A.6)then it follows that V ( µ, λ, φ ) = V tree ( µ ( t ) , λ ( t ) , φ ( t )) + subleading terms (A.7)Thus, with the choice of t in Eq. (A.6), the RG improved 1-loop effective potential reduces tothe tree-level potential with renormalized variables. Moreover, this choice of t manifests thefield dependent renormalization, i.e. φ → t ( φ ) → λ i ( φ ), through the implicit t -dependence.Since one can find the renormalization scale µ ( t ) corresponding to the t , the renormalizationof the couplings with respect to the renormalization scale leads to the renormalization of thecouplings with respect to the field value. – 20 – Perturbative solution of the renormalisation group equations
B.1 Zeroth order solution
In this Appendix, we solve Eq. (2.15) analytically neglecting colored terms to obtain Eq. (B.10).Substituting Eq. (2.12) into Eq. (2.15) gives ddx (cid:0) m L + m H u (cid:1) (0) = 12 ddx D (0) t + 163 g M − g M − g M (B.1)which can be solved as (cid:2) m L ( x ) + m H u ( x ) (cid:3) (0) = m L (0)+ m H u (0)+ 12 (cid:104) D (0) t ( x ) − D t (0) (cid:105) − (cid:90) x (cid:18) g M − g M − g M (cid:19) dt (B.2)To evaluate (B.2), it is enough to solve for D t . Solving Eqs. (2.6),(2.9) and (2.12) usingEq. (2.19),1 λ t = e (cid:82) x ( g +3 g + g ) dt (cid:20) λ t (0) − (cid:90) x e − (cid:82) t ( g +3 g + g ) dt (cid:48) dt (cid:21) (B.3) A t ( x ) (0) = e (cid:82) x λ t dt (cid:20) A t (0) + (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt (cid:21) (B.4) D t ( x ) (0) = e (cid:82) x λ t dt (cid:20) D t (0) + (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) λ t A t − g M − g M − g M dt (cid:19)(cid:21) (B.5)Using Eq. (B.4) and integrating by parts, (cid:90) x e − (cid:82) t λ t dt λ t A t dt = (cid:90) x λ t e (cid:82) t λ t dt (cid:48) (cid:20) A t (0) + (cid:90) t e − (cid:82) λ t dt (cid:48) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt (cid:21) (B.6)= − A t (0) + e (cid:82) x λ t dt (cid:20) A t (0) + (cid:90) t e − (cid:82) t λ t dt (cid:48) (cid:18) g M + 3 g M + 1315 g M dt (cid:19)(cid:21) − (cid:90) x (cid:18) g M + 3 g M + 1315 g M (cid:19) (cid:20) A t (0) + (cid:90) t e − (cid:82) t (cid:48) λ t dt (cid:48)(cid:48) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt (cid:48) (cid:21) dt (B.7)Using Eqs (2.20) to (2.21), (cid:90) (cid:18) g M + 3 g M + 1315 g M (cid:19) dx = − M ( x ) + 3 M ( x ) + 1399 M ( x ) (B.8)– 21 –nd hence Eq. (B.7) can be integrated by parts, (cid:90) x e − (cid:82) t λ t dt λ t A t dt = − A t (0) − A t (0) (cid:90) x (cid:18) g M + 3 g + 1315 g M (cid:19) dt + e (cid:82) λ t dt (cid:20) A t (0) + (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt (cid:21) + 2 (cid:18) M − M − M (cid:19) (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt − (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) M − M − M (cid:19) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt (B.9)Therefore, (cid:0) m L + m H u (cid:1) (0) = m L (0) + m H u (0) + 12 D t (0) (cid:16) e (cid:82) x λ t dt − (cid:17) − (cid:90) x (cid:18) g M − g M − g M (cid:19) dt − A t (0) e (cid:82) x λ t dt − A t (0) e (cid:82) x λ t dt (cid:90) x (cid:18) g M + 3 g M + 1315 g M (cid:19) dt − e (cid:82) x λ t dt (cid:90) x e − (cid:82) t λ t dt (cid:18) g M + 3 g M + 1315 g M (cid:19) dt − e (cid:82) x λ t dt (cid:90) x e − (cid:82) t λ t dt (cid:18) M − M − M (cid:19) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt + e (cid:82) x λ t dt (cid:18) M − M − M (cid:19) (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt + 12 e (cid:82) x λ t dt (cid:20) A t (0) + (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) g M + 3 g M + 1315 g M (cid:19) dt (cid:21) (B.10)= m L (0) + m H u (0) + (cid:88) X α (0) X X (0) with X ∈ { D t , A t , M i , M i M j | i, j = 1 , , } (B.11)where the coefficients α (0) X are functions of g (0) , g (0) , g (0) , λ t (0) and x . For example, α (0) A t M = 163 e (cid:82) x λ t dt (cid:90) x g (0)1 + 3 g (0) t dt + 133 e (cid:82) x λ t dt (cid:90) x e − (cid:82) t λ t dt (cid:48) g (0)1 + 3 g (0) t dt (B.12)where λ t is given by Eq. (B.3). We evaluated these integrals numerically, and the resultingnumerical values of α (0) i for tan β = 5 , , , ,
30 and AD scale = 10 , , GeV areprovided in Tables 5 to 9.
B.2 First order solution
Substituting Eqs. (2.12) to (2.14), Eq. (2.15) can be written as ddx (cid:0) m L + m H u (cid:1) (1) = 3361 ddx D (1) t − ddx D b + 1961 ddx D τ + 19261 g M + 14461 g M + 192305 g M (B.13)– 22 –nd solved as (cid:2) m L ( x ) + m H u ( x ) (cid:3) (1) = m L (0) + m H u (0)+ 3361 (cid:104) D (1) t ( x ) − D t (0) (cid:105) − (cid:104) D (1) b ( x ) − D b (0) (cid:105) + 1961 (cid:104) D (1) τ ( x ) − D τ (0) (cid:105) + (cid:90) x (cid:18) g M − g M + 192305 g M (cid:19) dt (B.14)Solutions of Eqs. (2.6) to (2.14) in the form of Eq. (2.19) are1 λ b = e (cid:82) x ( g +3 g + g − λ t ) dt (cid:20) λ b (0) − (cid:90) x e − (cid:82) t ( g +3 g + g − λ t ) dt (cid:48) dt (cid:21) (B.15) A b = e (cid:82) x λ b dt (cid:20) A b (0) + (cid:90) x e − (cid:82) t λ b dt (cid:48) (cid:18) λ t A t + 163 g M + 3 g M + 715 g M (cid:19) dt (cid:21) (B.16) D b = e (cid:82) x λ b dt (cid:20) D b (0) + (cid:90) x e − (cid:82) t λ b dt (cid:48) (cid:18) λ t A t + λ t D t + 6 λ b A b − g M − g M − g M (cid:19) dt (cid:21) (B.17)1 λ τ = e (cid:82) x ( g + g − λ b ) dt (cid:20) λ τ (0) − (cid:90) x e (cid:82) t ( g + g − λ b ) dt (cid:48) dt (cid:21) (B.18) A τ = e (cid:82) x λ τ dt (cid:20) A τ (0) + (cid:90) x e − (cid:82) t λ τ dt (cid:48) (cid:18) λ b A b − g M − g M (cid:19) dt (cid:21) (B.19) D τ = e (cid:82) x λ τ dt (cid:20) D τ (0) + (cid:90) x e − (cid:82) t λ τ dt (cid:48) (cid:18) λ b A b + 3 λ t D b + 4 λ τ A τ − g M − g M (cid:19) dt (cid:21) (B.20)1 λ t = e (cid:82) x ( g +3 g + g − λ b ) dt (cid:20) λ t (0) − (cid:90) x e (cid:82) t ( g +3 g + g − λ b ) dt (cid:48) dt (cid:21) (B.21) A (1) t = e (cid:82) x λ t dt (cid:20) A t (0) + (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:18) λ b A b + 163 g M + 3 g M + 1315 g M (cid:19) dt (cid:21) (B.22) D (1) t = e (cid:82) x λ t dt D t (0)+ e (cid:82) x λ t dt (cid:90) x e − (cid:82) t λ t dt (cid:48) (cid:20) λ b (cid:0) A b + D b (cid:1) + 6 λ t A t − g M − g M − g M (cid:21) dt (B.23)Substituting Eqs. (B.16),(B.19) and (B.22) into Eqs. (B.17), (B.20) and (B.23) respectively usingthe similar technique we used in Eqs. (B.8) and (B.9), one can express Eq. (B.14) in an integralform with a few integrals. This can be expressed as (cid:2) m L ( x ) + m H u ( x ) (cid:3) (1) = m L (0) + m H u (0) + (cid:88) X α (1) X X (0) (B.24)with X ∈ { D α , A α A β , A α M i , M i M j | α, β = t, b, τ and i, j = 1 , , } where the α (1) X are functions of g (0) , g (0) , g (0) , λ t (0) , λ b (0) , λ τ (0) and x . After numericallyevaluating the integrals in Eqs. (B.16), (B.19) and (B.22) and substituting the results into D b , D τ and D (1) t in Eq. (B.24), one can obtain the numerical values of α (1) i . The results are providedin Tables 5 to 9. – 23 – .3 Numerical coefficients of the baryogenesis condition The numerical coefficients in Eqs. (2.23) and (B.24) are given in the following tables.tan β = 5AD scale α (0) D t α (0) A t α (0) A t M α (0) A t M α (0) A t M α (0) M α (0) M α (0) M α (0) M M α (0) M M α (0) M M GeV 0.26 0.40 0.01 0.05 0.18 -0.05 -0.35 -0.08 0.00 0.01 0.0010 GeV 0.33 0.54 0.02 0.09 0.28 -0.07 -0.45 -0.11 0.00 0.03 0.0110 GeV 0.40 0.71 0.02 0.13 0.41 -0.09 -0.55 -0.13 0.00 0.05 0.01AD scale α (1) D t α (1) A t α (1) A t M α (1) A t M α (1) A t M α (1) M α (1) M α (1) M α (1) M M α (1) M M α (1) M M GeV 0.26 0.40 0.01 0.05 0.18 -0.05 -0.33 -0.08 0.01 0.02 0.0010 GeV 0.33 0.54 0.02 0.09 0.28 -0.07 -0.42 -0.11 0.02 0.03 0.0110 GeV 0.40 0.71 0.02 0.13 0.42 -0.09 -0.52 -0.13 0.03 0.05 0.01
Table 5 : Numerical coefficients of m L + m H u at zeroth and the first order for tan β = 5.FlexibleSUSY was used to get the low scale values of the couplings λ t (0) = 0 . , λ b (0) =0 . , λ τ (0) = 0 . , g (0) = 0 . , g (0) = 0 . , g (0) = 0 .
97 at m s = 6 TeV.tan β = 10AD scale α (0) D t α (0) A t α (0) A t M α (0) A t M α (0) A t M α (0) M α (0) M α (0) M α (0) M M α (0) M M α (0) M M GeV 0.38 0.01 0.05 0.17 0.25 -0.05 -0.35 -0.08 0.00 0.01 0.0010 GeV 0.51 0.01 0.08 0.27 0.31 -0.07 -0.45 -0.11 0.00 0.03 0.0010 GeV 0.66 0.02 0.13 0.39 0.38 -0.09 -0.55 -0.13 0.00 0.05 0.01AD scale α (1) D t α (1) A t α (1) A t M α (1) A t M α (1) A t M α (1) M α (1) M α (1) M α (1) M M α (1) M M α (1) M M GeV 0.38 0.01 0.05 0.17 0.25 -0.05 -0.33 -0.08 0.01 0.01 0.0010 GeV 0.51 0.01 0.08 0.27 0.32 -0.07 -0.42 -0.11 0.02 0.03 0.0110 GeV 0.67 0.02 0.13 0.39 0.38 -0.09 -0.52 -0.13 0.03 0.05 0.01
Table 6 : Numerical coefficients of m L + m H u at zeroth and first order for tan β = 10. Used λ t (0) = 0 . , λ b (0) = 0 . , λ τ (0) = 0 . , g (0) = 0 . , g (0) = 0 . , g (0) = 0 .
97 at m s = 6 TeV.– 24 –an β = 15AD scale α (0) D t α (0) A t α (0) A t M α (0) A t M α (0) A t M α (0) M α (0) M α (0) M α (0) M M α (0) M M α (0) M M GeV 0.36 0.01 0.05 0.16 0.24 -0.05 -0.35 -0.08 0.00 0.01 0.0010 GeV 0.48 0.01 0.08 0.25 0.30 -0.07 -0.44 -0.10 0.00 0.03 0.0010 GeV 0.62 0.02 0.12 0.37 0.36 -0.09 -0.55 -0.13 0.00 0.04 0.01AD scale α (1) D t α (1) A t α (1) A t M α (1) A t M α (1) A t M α (1) M α (1) M α (1) M α (1) M M α (1) M M α (1) M M GeV 0.36 0.01 0.05 0.16 0.24 -0.05 -0.33 -0.08 0.01 0.01 0.0010 GeV 0.49 0.01 0.08 0.25 0.30 -0.07 -0.42 -0.10 0.02 0.03 0.0010 GeV 0.63 0.02 0.12 0.37 0.36 -0.09 -0.51 -0.13 0.03 0.04 0.01
Table 7 : Numerical coefficients of m L + m H u at zeroth and first order for tan β = 15. Used λ t (0) = 0 . , λ b (0) = 0 . , λ τ (0) = 0 . , g (0) = 0 . , g (0) = 0 . , g (0) = 0 .
97 at m s = 6 TeV.tan β = 20AD scale α (0) D t α (0) A t α (0) A t M α (0) A t M α (0) A t M α (0) M α (0) M α (0) M α (0) M M α (0) M M α (0) M M GeV 0.36 0.01 0.05 0.16 0.24 -0.05 -0.35 -0.08 0.00 0.01 0.0010 GeV 0.49 0.01 0.08 0.25 0.30 -0.07 -0.44 -0.10 0.00 0.03 0.0010 GeV 0.63 0.02 0.12 0.37 0.37 -0.09 -0.55 -0.13 0.00 0.04 0.01AD scale α (1) D t α (1) A t α (1) A t M α (1) A t M α (1) A t M α (1) M α (1) M α (1) M α (1) M M α (1) M M α (1) M M GeV 0.37 0.01 0.05 0.16 0.25 -0.05 -0.33 -0.08 0.01 0.01 0.0010 GeV 0.49 0.01 0.08 0.26 0.31 -0.07 -0.42 -0.10 0.02 0.03 0.0110 GeV 0.64 0.02 0.12 0.37 0.37 -0.09 -0.51 -0.13 0.03 0.05 0.01 α (1) A b M α (1) A τ α (1) D τ α (1) A τ M GeV 0.00 0.01 0.01 0.0010 GeV 0.01 0.01 0.01 0.0010 GeV 0.01 0.01 0.01 0.01
Table 8 : Numerical coefficients of m L + m H u at zeroth and first order for tan β = 20. Used λ t (0) = 0 . , λ b (0) = 0 . , λ τ (0) = 0 . , g (0) = 0 . , g (0) = 0 . , g (0) = 0 .
96 at m s = 6 TeV.– 25 –an β = 30AD scale α (0) D t α (0) A t α (0) A t M α (0) A t M α (0) A t M α (0) M α (0) M α (0) M α (0) M M α (0) M M α (0) M M GeV 0.36 0.01 0.05 0.16 0.24 -0.05 -0.35 -0.08 0.00 0.01 0.0010 GeV 0.49 0.01 0.08 0.25 0.30 -0.07 -0.44 -0.10 0.00 0.03 0.0010 GeV 0.63 0.02 0.12 0.37 0.37 -0.09 -0.55 -0.13 0.00 0.04 0.01AD scale α (1) D t α (1) A t α (1) A t M α (1) A t M α (1) A t M α (1) M α (1) M α (1) M α (1) M M α (1) M M α (1) M M GeV 0.37 0.01 0.05 0.16 0.25 -0.05 -0.33 -0.08 0.01 0.02 0.0010 GeV 0.50 0.01 0.08 0.26 0.31 -0.07 -0.42 -0.10 0.02 0.03 0.0110 GeV 0.65 0.02 0.12 0.38 0.37 -0.09 -0.51 -0.13 0.04 0.05 0.01 α (1) A b α (1) A b A t α (1) A b M α (1) A b M α (1) A b M α (1) A τ α (1) D τ α (1) A τ M α (1) A τ M GeV 0.00 0.00 0.00 0.01 0.00 0.01 0.01 0.00 0.0110 GeV 0.00 0.01 0.00 0.01 0.00 0.02 0.02 0.00 0.0110 GeV 0.01 0.01 0.01 0.02 0.01 0.02 0.02 0.01 0.02
Table 9 : Numerical coefficients of m L + m H u at zeroth and first order for tan β = 30. Used λ t (0) = 0 . , λ b (0) = 0 . , λ τ (0) = 0 . , g (0) = 0 . , g (0) = 0 . , g (0) = 0 .
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