Affleck-Dine leptogenesis and its backreaction to inflaton dynamics

We investigate the backreaction of the Affleck-Dine leptogenesis to inflaton dynamics in the F-term hybrid and chaotic inflation models in supergravity. We determine the lightest neutrino mass in both models so that the predictions of spectral index, tensor-to-scalar ratio, and baryon abundance are consistent with observations.


Introduction
The success of the Big Bang nucleosynthesis theory requires a large amount of baryon asymmetry at least at the temperature of 1 MeV in the early Universe. However, the baryon asymmetry must be washed out due to the primordial inflation, so that there has to be some mechanism to generate the baryon asymmetry after inflation. The Affleck-Dine baryo/leptogenesis is a promising candidate of baryogenesis in supersymmetric (SUSY) theories [1][2][3]. A B − L charged scalar field with a flat potential, called an Affleck-Dine (AD) field, can obtain a large tachyonic effective mass and have a large vacuum expectation value (VEV) during and after inflation. As the energy of the Universe decreases, the effective mass becomes inefficient and the AD field starts to oscillate coherently around the origin of its potential. At the same time, the phase direction of the AD field is kicked by its A-term potential. Since the B − L number density is proportional to the phase velocity of the AD field, the B − L asymmetry is generated through this dynamics. Finally, the coherent oscillation decays and dissipates into thermal plasma and then the B − L asymmetry is converted to the desired baryon asymmetry through the sphaleron effects [4,5].
Since the AD field obtains a large VEV during inflation, we should take into account its effect on inflaton dynamics via supergravity effects. 1 In fact, there are many works revealing that a constant term in superpotential and a scalar field with a large VEV may affect inflaton dynamics [10][11][12][13]. These effects may rescue the * Correspondence to: ICRR, The University of Tokyo, Kashiwa, Chiba 277-8582, Japan.
E-mail address: yamadam@icrr.u-tokyo.ac.jp. 1 See Refs. [6,7] for the case that the AD field also plays the role of inflaton. See also Refs. [8,9] for related works in non-SUSY models.
F -term hybrid and chaotic inflation models, which themselves are somewhat inconsistent with the observations of CMB temperature fluctuations. In this letter, we apply their calculation to the scenario of the Affleck-Dine leptogenesis, focusing on the L H u flat direction in the minimal SUSY standard model sector. We show that the backreaction of the AD field on the inflaton dynamics can rescue the F -term hybrid and chaotic inflation models and the baryon asymmetry can be consistent with the observation at the same time. We predict extremely small mass for the lightest neutrino, which allows us to calculate the effective Majorana mass for the 0νββ decay process.

Potential of the AD field
Let us focus on the dynamics of the L H u flat direction: where L i and H u are left-handed slepton with a family index i and up-type Higgs, respectively. Since the observations of neutrino oscillation imply nonzero masses of neutrinos, we introduce a superpotential of where H u = sin β ×174 GeV and tan β ≡ H u / H d . We take the mass basis where the mass matrix for the neutrinos is diagonal.
Here, the flat direction corresponding to the lightest neutrino is most important for the purpose of the Affleck-Dine leptogenesis, so that we identify that direction as the AD field and denote it as φ.
The coupling constant λ is determined to account for the observations of baryon asymmetry and CMB temperature fluctuations.
The relevant potential of the AD field φ is written as is the F -term potential. The potential V soft represents the Higgs μ term and soft SUSY breaking terms in low energy: where m φ (= O(1) TeV) is the mass of the L H u flat direction and m 3/2 is gravitino mass. We expect that the coefficient of A-term a is of order unity. The potential of V H is a so-called Hubble-induced mass term, which comes from supergravity effects [3]. In supergravity, the potential of scalar fields is given by where K is a Kähler potential and Pl . The subscripts represent derivatives with respect to corresponding fields and K i¯j ≡ (K i¯j ) −1 . In order to realize the Affleck-Dine leptogenesis, we assume where S is an inflaton superfield and c is an O (1) constant. When we consider F -term inflation models, where the F -term of inflaton Pl with H inf being the Hubble parameter during inflation, we obtain a Hubble-induced mass of the AD field: To realize the Affleck-Dine leptogenesis, we assume c H < 0.
After inflation ends, the inflaton gradually decays into radiation and a background plasma develops with a temperature of where g * (T ) is the effective number of relativistic degrees of freedom in the thermal plasma. The decay rate of inflaton I is related with the reheating temperature T RH as T RH 90 The AD field acquires a thermal potential via 2-loop effect when its VEV is larger than the temperature: where c T = 45/32 and α s ≡ g 2 s /4π is a strong coupling constant [14,15].

Dynamics of the AD field
When we consider F -term inflation models, the Hubbleinduced mass term of Eq. (9) arises during inflation. Since we assume c H < 0, the AD field stays at the following potential minimum: After inflation ends, its VEV is determined as Eq. (14) with the replacement of H inf → H(t) during the inflaton oscillation dominated era. Note that the phase direction of the AD field stays at a certain phase due to the Hubble friction effect. When the Hubble parameter decreases to m φ or (φ −1 V T ) 1/2 , the AD field starts to oscillate around the origin of the potential. We denote the Hubble parameter as H osc : where φ osc is the VEV of the AD field at the beginning of oscillation. At the same time, the AD field starts to rotate in the complex plane because its phase direction is kicked by the A-term of Eq. (6).
where θ 0 is an initial phase of the AD field. Here, we define the ellipticity parameter (≤ 1), which represents the efficiency of the Affleck-Dine mechanism. Finally, the coherently oscillating AD field decays and dissipates into thermal plasma [16] and the sphaleron effect converts the B − L asymmetry to baryon asymmetry [4,5]. The resulting baryonto-entropy ratio Y b is given by where 8/23 in the first line is the sphaleron factor [17]. In the second line, we assume α s (15). Note that the result is independent of the reheating temperature [15] once we satisfy 2 2 One may wonder that such a high reheating temperature is excluded by the gravitino problem. This is the case of unstable gravitinos with mass of m 3/2 = O(100) GeV [18]. In this letter, we assume gravitino to be stable. In this case, the next to lightest SUSY particle may decay in the epoch of Big Bang nucleosynthesis and may destroy light elements. This problem can be avoided when sneutrino is the next to lightest SUSY particle [18]. This constraint is highly model dependent, so that in this letter we use a conservative bound such that the gravitino abundance is below the observed dark matter (DM) abundance [19]. Note that the gravitino abundance can be consistent with the observed DM abundance when we take an appropriate reheating temperature (e.g., T RH ∼ 10 10 GeV for m 3/2 = 100 GeV [20]).
where we assume α s ≈ 0.1. The observed baryon asymmetry of Y obs b 8.6 × 10 −11 [21] can be explained when the coupling λ sat- Note that our calculations can be applied to gauge-mediated SUSY breaking models as well as gravity-mediated ones because the mass of L H u flat direction comes from the Higgs μ-term. The thermal effect of Eq. (13) is also model-independent. Thus we can consider the case that gravitino mass is much smaller than the electroweak scale.

F -term hybrid inflation
In this subsection, we consider the simplest model of F -term hybrid inflation [22,23] taking into account the effect of the AD field on the dynamics of inflaton. The superpotential of the inflaton sector is given by where κ is a coupling constant, S is inflaton, and ψ and ψ are waterfall fields. When the inflaton S has a sufficiently large VEV, the waterfall fields obtain large effective masses of κ S and thus stay at the origin of the potential. The inflaton S obtains the Coleman-Weinberg potential of where S cr ≡ v/ √ 2. The inflaton S slowly rolls down to the origin of the potential until its VEV reaches the critical value of S cr . During this slow roll, the energy density is dominated by the F -term potential energy of κ 2 v 4 /4, so that inflation occurs. When the inflaton S reaches a critical VEV of S cr , the waterfall fields and inflaton start to oscillate about their global minimum and inflation ends. In this simplest model, the spectral index is predicted as n s 1 − 1/N * 0.98, where N * (≈ 55) is the e-folding number at the horizon exit of the CMB scale. This prediction is inconsistent with the observed value more than 2 sigma level: n (obs) s = 0.963 ± 0.008 [24]. Now we take into account the backreaction of the AD field to the dynamics of the inflaton. In supergravity, the potential of scalar fields is determined by Eq. (7). When we consider the total superpotential W = W (AD) + W (inf) , the terms of W S KS φ Wφ + c.c. − 3 |W | 2 give a linear potential of inflaton such as [11] V BR a where a is an O (1) constant determined by higher-dimensional Kähler potentials and W (AD) is determined by Eqs. (2) and (14).
Hereafter we assume a = 1.
The effect of the linear term in the F -term hybrid inflation model has been studied in Ref. [11]. They have found that the linear term affect the inflaton dynamics when the slope of the linear term is the same order with that of the Coleman-Weinberg potential. They introduce a parameter to describe the relative importance of the two contributions to the slope: ξ ≡ 2 9/2 π 2 κ 3 ln 2 which should be smaller than unity so that the inflaton can rolls towards the critical value. When ξ is of order but below unity, the linear term is efficient for the inflaton dynamics. We define a critical value of coupling constant for the AD field such as where we use H 2 inf = κ 2 v 4 /12M 2 Pl . When λ is near the critical value, ξ is close to unity and the backreaction of the AD field to inflaton dynamics is efficient. Note that λ should be not larger than λ c so that the inflaton can roll towards the critical value and inflation can end.
Since the linear term breaks R-symmetry, under which the inflaton S is charged, we need to investigate the inflaton dynamics in its complex plane as done in Ref. [11]. 3 We read their result of Fig. 9, where desired values of W (AD) can be read from the contours of gravitino masses by the relation of m 3/2 M 2 The result is shown in Fig. 1, where the spectral index as well as the baryon asymmetry can be consistent with the observed values in the colored region. Here, we assume that the final phase of the inflaton is larger than π/32 to avoid a fine-tuning of initial condition. Above the red-dashed curves for each case of gravitino mass, we can neglect the effect of a linear term arising from low energy SUSY breaking, which is investigated in the original work of Ref. [11]. If there is only the effect of a linear term arising from low energy SUSY breaking, the spectral index can be consistent with the observation on the red-dashed curve for each case of gravitino mass. Thus we can explain the observation on and above the red-dashed curve for each case of gravitino mass in our model. 5 3 A CP-odd component of inflaton is excited via this dynamics, which also provides another scenario of baryogenesis [25]. 4 The dynamics of the phase direction of the AD field can be neglected for the case of λ κ [26], which is actually satisfied in our case, so that the dynamics of inflaton is basically equivalent to the one in Ref. [11].
The right region is excluded by the cosmic string bound [24]. We set reheating temperature such that the gravitino abundance generated from inflaton decay [27] and scattering in the thermal plasma [28,29] is minimized, where we check that the reheating temperature satisfies the condition of Eq. (21). The upper-right regions (above the upper green dot-dashed line for m 3/2 ≤ 100 GeV and the lower one for m 3/2 = 1 TeV) are excluded by the overproduction of gravitinos if they are stable. Note that if gravitino is unstable, the bound is much severer than the case of stable gravitino. For the cases of stable gravitino with mass heavier than a few TeV, we find that there is no viable region because of the gravitino problem [11].
Since the value of superpotential of the AD field is determined at each point in Fig. 1, we can determine its coupling constant λ.
Then we can use Eq. (20) to calculate the baryon abundance. For the cases of m 3/2 = 100 GeV and 1 TeV, we can explain the baryon abundance by taking ˜ properly. On the other hand, for the case of m 3/2 = 100 MeV, the baryon asymmetry cannot be produced efficiently below the blue-doted curve even if ˜ is as large as unity. 6 We predict lightest neutrino mass m ν as given in the contour plot.
Since the coupling constant in the superpotential of the AD field is roughly determined by Eq. (27)

Chaotic inflation
In this subsection, we consider a chaotic inflation model in supergravity where an inflaton superfield I has Z 2 and shift symmetries in the Kähler potential [30]: The imaginary part of its scalar component η ≡ (I − I * )/ √ 2 is identified with inflaton. The shift symmetry is explicitly broken by a superpotential of where S is a stabilizer field with a Kähler potential of Eq. (8). When the inflaton has a large VEV, the stabilizer field obtains a large effective mass and stays at the origin. The inflaton potential is then given by the quadratic potential via the F -term of the stabilizer field. Thanks to the shift symmetry in the Kähler potential, the VEV of inflaton can be larger than the Planck scale and quadratic chaotic inflation can be realized in this model. Note that reheating can occur when we introduce the interaction term of y X H u H d in the superpotential [30]. We can avoid the gravitino problem to set the coupling constant y appropriately (see also footnote 2).
Here, we take into account the backreaction of the AD field. The full supergravity potential is given by where c is the parameter in the Kähler potential [see Eq. (8)]. This potential implies that the effect of the AD field is relevant when its 6 When the coefficient of A-term a in Eq. (6) is much larger than unity, ˜ can be larger than unity and the bound of the blue curve disappear. Fig. 2. Spectral index and tensor-to-scalar ratio in the chaotic inflation model with the backreaction of the AD field. The red, green, and blue dots represent our results at e-folding numbers of 50, 55, and 60, respectively. We randomly take 100 points for the parameters c and λ within the intervals of [1,10] and [0, 100m/M Pl ], respectively. We plot the results as the light dots for the case of λ/λ c < 0.5, 5 < λ/λ c , or c < 5. The blue regions are the 1σ (deep colored regions) and 2σ (pale colored regions) constraints of the Planck experiment [31]. For comparison with standard results, we plot the predictions in the chaotic inflation models with linear and quadratic potentials without the backreaction as the black thin and thick lines, respectively, where the results are given as intersection points of black lines and dashed lines for corresponding e-folding numbers. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) VEV is as large as the Planck scale. Since H inf ∼ 10m in the chaotic inflation model, the VEV of the AD field is as large as the Planck scale for the case of [see Eq. (14)]. We numerically solve the equations of motion of the inflaton η and the AD field φ and calculate the tensor-to-scalar ratio and spectral index. We show the result in Fig. 2, where we take the parameters c and λ randomly within the intervals of [1,10] and [0, 100m/M Pl ], respectively. The red, green, and blue dots represent the results at e-folding numbers of 50, 55, and 60, respectively. As a result, the tensor-to-scalar ratio can be as small as 0.14, 0.13, and 0.12 at the e-folding number of 50, 55, and 60, respectively, which is marginally consistent with the present upper bound within 2σ . We plot the results as the light dots for the case of λ/λ c < 0.5, 5 < λ/λ c , or c < 5, which clarifies that the tensor-toscalar ratio can be smaller only for the case of 0.5 < λ/λ c < 5 and c > 5. This requires that the coupling constant in the superpotential is of order 10m/M Pl ∼ 10 −4 , so that the lightest neutrino mass is predicted to be of order 10 −9 eV. Note that the resulting baryon asymmetry of Eq. (20) is naturally consistent with the observation when gravitino mass is of order 100 GeV-1 TeV. Finally, we also perform numerical calculations including higher-dimensional Kähler potentials of and find that the tensor-to-scalar ratio cannot be smaller than about 0.11 at the e-folding number of 60 even in this case. 7 This is in contrast with the result of Ref. [13], where they have investigated the effect of an additional scalar field to chaotic inflation in a non-SUSY model and found that the tensor-to-scalar ratio can be much smaller than 0.1. This is because the exponential factor in the supergravity potential of Eq. (30) (or, precisely speaking, the exponential dependence on Kähler potential of scalar potential) makes the VEV of the AD field smaller and its backreaction to the inflaton dynamics smaller in supergravity.

Discussion and conclusions
We have investigated the backreaction of the AD field to inflaton dynamics, focusing on the L H u flat direction in the minimal SUSY standard model. In the F -term hybrid inflation model, a linear term of inflaton potential is induced by the nonzero superpotential of the AD field. As a result, the spectral index as well as baryon abundance can be consistent with the observed values. In the chaotic inflation model with a shift symmetry in the inflaton Kähler potential, we have found that the tensor-to-scalar ratio can be as small as 0.12 due to the backreaction of the AD field.
All of the above scenarios require a large VEV of the AD field during inflation. This is also favored in light of avoiding the baryonic isocurvature constraint, which is particularly important in the chaotic inflation model [32][33][34][35]. To realize a large VEV during inflation for the L H u flat direction, the mass of the lightest neutrino has to be extremely small. Thus the total neutrino mass is given by m ν 0.06 eV for NH 0.1 eV for IH, for the cases of normal hierarchy (NH) and inverted hierarchy (IH), respectively. We can also calculate the upper and lower bounds on the effective Majorana mass for the 0νββ decay process such as [15,36] 0.001 eV m ββ 0.004 eV for NH (34) 0.01 eV m ββ 0.04 eV for IH, where we take the values for the experimentally measured parameters from Ref. [37]. These results of total neutrino mass and effective Majorana mass are too small to measure in the near future at least for the case of NH. Therefore, if we would measure the total neutrino mass or the effective Majorana mass in the near future, we can falsify our scenario of the AD leptogenesis.
On the other hand, if we would experimentally obtain only their upper bound and if the tensor-to-scalar ratio would be measured as 0.12-0.15, our scenario of the AD leptogenesis after the chaotic inflation would be more attractive.
Finally, let us comment on other scenarios of Affleck-Dine baryogenesis using other flat directions, such as the u c d c d c flat direction, where u c and d c are up-type and down-type right-handed squarks, respectively. In this case, there are some corrections in calculations of baryon abundance. The most important difference from our scenario is the possibility of the formation of nontopological solitons called Q-balls [38][39][40][41][42]. In particular, as we have shown in this letter, the backreaction of the AD field is relevant when its VEV is sufficiently large during inflation, which implies that large Q-balls may form after the AD baryogenesis. In this case, Q-balls may decay after DM (the lightest SUSY particle) freezes out, so that their decay can be a non-thermal source of DM. There are interesting scenarios that the non-thermal production of DM from Q-ball decay can naturally explain the coincidence between the energy density of baryon and DM [43][44][45][46] (see, e.g., Ref. [34] in the case of chaotic inflation).