More on cosmological constraints on spontaneous R-symmetry breaking models

We study the spontaneous R-symmetry breaking model and investigate the cosmological constraints on this model due to the pseudo Nambu-Goldstone boson, R-axion. We consider the R-axion which has relatively heavy mass in order to complement our previous work. In this regime, model parameters, R-axions mass and R-symmetry breaking scale, are constrained by Big Bang Nucleosynthesis and overproduction of the gravitino produced from R-axion decay and thermal plasma. We find that the allowed parameter space is very small for high reheating temperature. For low reheating temperature, the U(1)_R breaking scale f_a is constrained as f_a<10^{12-14}GeV regardless of the value of R-axion mass.


Introduction
Supersymmetry (SUSY) is one of the most promising candidates of the physics beyond the standard model (SM) because it can relax the naturalness problem and suggests the gauge coupling unification. It is also widely believed to be one of the key ingredients for constructing a consistent string theory encompassing the SM. Since SUSY has not been observed in experiments yet, it has to be broken somewhere between the weak scale and the Planck scale.
GeV, the R-axion can efficiently decay into various hadrons. For sufficiently heavy R-axions, they immediately turn to hadronic jets, which affects successful BBN. This phenomenon can constrain the parameter space of R-axions. The other constraint comes from gravitino production. In this short note, we mainly focus on the regime in which gauge mediation is the dominant contribution to the mediation of SUSY breaking. In this case, the gravitino is a stable particle and can be over-produced via R-axion decay. In fact, as we will show below, overproduction of such gravitinos yields a condition which is complementary to the one for thermal production of gravitino.
In this paper, we investigate cosmological effects of heavy R-axion whose mass scale is not covered in the previous work [8]. Especially, we focus the mass scale heavier than 3 GeV to avoid subtlety of non-relativistic decay into pions which requires careful treatment because it does not necessarily destroy the light elements constructed by BBN. Also, we assume that all superparticles (except gravitino) are heavier than the R-axion because R-axion decay into superparticles requires highly model dependent argument. If the decay channel into superparticles of R-axions opens, the constraint would become more severe. The organization of this paper is as follows: In section 2, we briefly review the R-symmetry breaking model. Then, we examine the lifetime of the R-axion and the branching ratio of hadronic decays. In section 3, we firstly review the mechanisms of the R-axion production and its abundance. Then we impose constraints on R-axion abundance from the BBN and gravitino overproduction. As we will show that the R-axion mass and R-symmetry breaking scale are severely constrained. Section 4 is devoted to summary and discussion.

Hadronic decay of R-axion
In this section, we briefly review the simple model with spontaneously broken R-symmetry studied in Ref. [8]. Let us focus on the R-charged light SUSY-breaking field, X, and consider a low-energy scale where all other fields including messenger fields are integrated out. The effective superpotential is, then, assumed to be Here the constant term W 0 is required to realize vanishing cosmological constant. Note that this class of models is common in various F -term supersymmetry breaking models [4]. We assume non-canonical Kähler potential yielding the following potential, which realizes a potential minimum with spontaneously broken R-symmetry. Here M pl is the reduced Planck mass. The first term comes from our assumption about Kähler potential. f a is turned out to be the "axion decay constant" and we assume f a ≪ M pl . The second term appears from the Planck-suppressed interaction in supergravity theory. It breaks R-symmetry explicitly and generates the mass term for the R-axion, the phase component of the X field.
In coupling to the gravity, vanishing cosmological constant requires Though we have introduced an ad-hoc Kähler potential for the SUSY-braking vacuum with spontaneously broken R-symmetry, we believe that it is a simple toy model which reveals various aspects in spontaneous R-symmetry breaking models 2 [4]. Dividing X into VEV and fluctuation, we have the potential for s and a as Hereafter we call s R-saxion and a R-axion. From the potential, we can read off the R-axion mass m a and R-saxion mass m s as We expect that R-symmetry breaking potential Eq.(2.2) would be related to SUSY breaking and With this assumption, we find the relation of m a and m s as Meanwhile the gravitino mass m 3/2 is written by Since the gravitino is lighter than R-axion while R-saxion is heavier than R-axion, R-axion can decay into gravitinos but not R-saxions. Now we evaluate the interactions of R-axions, which is necessary to investigate the production and decay rate of R-axions. First, let us consider the interaction with the gauge fields.
R-axions couple with the SM gauge fields through anomaly couplings, where G i for i = 3, 2 and 1 represent the SM gauge groups SU(3) C , SU(2) L and U(1) Y , respectively and g i s are the corresponding gauge couplings. From these couplings, we obtain the decay rates of R-axion to each pair of gauge bosons as 11) where θ w is the Weinberg angle and m Z and m W are the Z-boson and W-boson masses, respectively. Since the anomaly coefficients are model-dependent parameters of the order of the unity, we take C i = 1 for all G i in the following. Taking other values of the order of the unity does not change our results significantly.
Secondly, the R-axion can also couple with the SM fermions through the mixing between the R-axion and the Higgs bosons [12]. Couplings with up type quarks, down type quarks, charged leptons and neutrinos are expressed as the effective interactions, λ f af γ 5 f, with the coupling constants GeV and m f denotes the mass of each fermion f . tan β is the ratio of the vacuum expectation values of the up-type Higgs boson H u and the down-type Higgs boson H d . From these couplings, R-axions can decay into a pair of fermions with the decay rates, Note that tan β as well as other parameters such as the stop mass determines the Higgs mass.
For the 125 GeV Higgs, tan β 10 is favored for the stop mass with a few TeV [2]. Finally, R-axions can decay to a pair of gravitinos through supergravity effect, Decay rate is given by for m a ≫ m 3/2 . Here we have used the relation Λ 2 eff = 3 1/4 f a M pl /2 m a and taken into account that for the light gravitino, the decay into the spin 1/2 goldstino component ψ with ψ µ ∼ i 2/3 ∂ µ ψ/m 3/2 dominates over the decay into the spin 3/2 component [13].
We plot the lifetime τ a of the R-axion for tan β = 30 in Fig.1. For 3GeV m a 8GeV, decay into tau pairs dominates; for 8GeV m a 100GeV, decay into bottom pairs dominates; and for m a 100GeV, decay into gluon or gravitino pairs dominates. Note that for large f a , the decaying into the gravitino pairs dominates other channels for the heavy R-axion mass since the suppression factor f a /M pl is not so small that overwhelms the loop factors. From this figure, We can see that the lifetime is shorter than 10 13 s for m a 3 GeV.
In the case that the energy injected quarks are high enough, right after the axion decay, quarks and gluons immediately turn into hadronic jets 3 . Hence, the process does not depend on the first particles created by axion decay. For the BBN constraint, then, only the branching ratio decaying into hadronic particles determines the constraint. We present the hadronic branching ratio of the R-axion B h for tan β = 30 in Fig.2, where B h is the sum of the branching ratio of the R-axion to colored particles. B h becomes small if we take larger f a since the branching ratio of the R-axion decay into gravitinos becomes sizable. It is 10GeV m a 10 2 GeV, decay channel into bottom pairs dominates the total decaying ratio, we have B h ∼ 1 regardless of f a . For 3GeV m a 10 GeV, B h becomes small once more because the decay channel to taus dominates the total decay ratio. In this range B h is of O(10 −2 ). Summarizing the above, B h is bigger than 10 −2 for m a 3 GeV, f a 10 16 GeV.
Thus, the constraint with B h = 1 gives the stringent constraint whereas that with B h = 10 −3 gives the conservative one, as we will see in the next section. 3 Cosmological constraints on R-axion abundance

R-axion production
Let us consider the R-axion cosmology. First we study the R-axion production. The R-axion production depends on cosmological scenarios, and we here simply suppose that the U(1) R breaking occurs after inflation. In this case, cosmic R-string forms at the time of the phase transition [9]. The cosmic R-strings enter the scaling regime quickly, and the cosmic string loops that are produced continuously emit R-axions, which is the first source of R-axions [14].
Gradually the explicit U(1) R breaking effect can no longer be neglected, and string network If we take f a < 10 13 GeV, we obtain the value of B h which is the same as one for f a = 10 13 GeV.
turns into string-wall system, which is unstable and decay to R-axions immediately [15]. This is the second source of R-axions. At the same time, coherent oscillation of R-axions also starts, which is the third source of R-axions [16]. In addition to these R-axion production from the dynamics, there are thermal production [17] and production from R-saxion decay of R-axions. Note that it can be shown that production from R-saxion decay is negligible (see Appendix of Ref. [8]). Here we summarize the R-axion production from the dynamics (the coherent oscillation, the decay of cosmic string, decay of the string-wall system) and thermal bath referring the result of Ref. [8]. The R-axion abundance produced from dynamics is given by where T R is reheating temperature. H R and H osc are Hubble parameters evaluated at the time of reheating and of beginning of the R-axion oscillation, respectively. Note that the result changes only by numerical factors, but parameter dependence does not change if we consider the scenario without cosmic string formation but only with the coherent oscillation of R-axion [8]. The abundance of R-axions produced thermally is given by where T D = 10 6 GeVg −6 3 C −2 3 (f a /10 10 GeV) 2 is the decoupling temperature. This value is fixed when R-axions become non relativistic. Note that for T R > T D , R-axions are once thermalized and the R-axion abundance becomes independent of the reheating temperature.

Constraints for parameter space
The standard BBN scenario can explain the light elements in the present Universe elegantly.

(3.4)
Since we have seen B h 10 −2 from Fig. 2, we obtain the conservative bound if we use (3.3). The larger B h is, the more severe constraint becomes. Then, (3.4) for B h = 1 gives more severe bound than (3.3). Note that B h = 1 is the good approximation for m a 10 GeV and f a 10 13 GeV. Later, we will show the results of both cases.
There is another cosmological constraint on the R-axion abundance. Since R-axions can decay into the stable gravitinos, their abundance from R-axion decay may overwhelm the present abundance of dark matter (DM). Since the gravitino abundance should not exceed that of DM, we have the following constraint as 3.5) where Br ψ is branching ratio of gravitino. Note that the present gravitino abundance has a suppression factor Br ψ (2m 3/2 /m a ), since a R-axion decays into two relativistic gravitinos with the total energy m a and gradually becomes nonrelativistic. Since this constraint comes from the present Universe, it exists regardless of the lifetime of R-axion if τ a < τ 0 . We should also note the constraint from the gravitino produced thermally. The thermally produced gravitino abundance is given by [21] which must be smaller than, again, 4.7 × 10 −10 GeV(Ω m h 2 /0.13), where mg is the gluino mass. Therefore, we need relatively large gravitino mass to avoid the overclosure problem, depending on the gravitino mass. In other words, relatively large f a and m a are required, see Eq. (2.9). In Fig. 3 and Fig. 4, we present the cosmological constraints on model parameters discussed above, in terms of m a and f a with various cases of the reheating temperature. Here we have set mg = 10TeV. If the reheating temperature is high, T R ≃ 10 6 GeV, thermal production of R-axion is so large that the gravitino abundance from the R-axion decay exceeds that of DM for m a 10 3 GeV and f a 10 9 GeV. Since the gravitinos produced thermally overclose the Universe for light gravitinos, which means small m a and f a , the allowed parameter space is very small. Allowed parameter region enlarges for lower reheating temperature. In particular, the gravitino overclosure problem is almost absent for T R < 1 GeV. In this case, model parameters are constrained only by the BBN. We can see that R-symmetry breaking scale f a is constrained from above regardless of the reheating temperature, f a 10 12−14 GeV. Note that we find that the constraint would not change so much if we use the precise value of B h (between 10 −3 to 1) by comparing the constraints of B h = 1 and 10 −3 .
We comment on the generality of the constraints shown in Fig. 3 and Fig. 4. R-axions couple with SM gauge fields through anomaly couplings and couple with the SM fermions through the mixing between the R-axion and the Higgs bosons which comes from B-term in Higgs potential. These couplings are general in spontaneous R-symmetry breaking models up to numerical factor. On the other hand, it would be possible to change the gravitino mass relation Eq. (2.9) and the R-axion gravitino coupling Eq. (2.15) if we consider more complicated superpotential. In order to avoid the constraint from gravitinos, we need a model to change Eq. (2.9) or Eq. (2.15).
It would be interesting to further explore heavier R-axion such as m a 10TeV. In this case, various decay modes into superparticles are open. Thus, arguments become highly model dependent. We will study some examples elsewhere.

Summary and discussions
In this paper, we considered the spontaneous R-symmetry breaking model and investigated the cosmological constraints of heavy R-axion which can decay to hadrons. This work complements our previous one [8]. We estimated the abundance of the R-axion produced by decay of R-string and domain wall, vacuum misalignment and thermal plasma. Such abundance is constrained by BBN and the gravitino overproduction. We showed cosmological constraints on model parameters, R-axion mass and R-symmetry breaking scale. As a result, we found that U(1) R breaking scale is constrained as f a < 10 12−14 GeV for low reheating temperature regardless of the value of R-axion mass. For high reheating temperature, BBN, gravitinos from R-axion decay, and gravitinos from thermal plasma constrain different parameter regions and the allowed parameter space is very small. In conclusion, even in the heavy R-axion regime, it has poor compatibility with relatively high reheating temperature T R 10 6 GeV. The constraints we showed in this paper can apply to many spontaneous R-symmetry breaking models and are important for phenomenological model building.
Finially it would be useful to re-interpret out results shown above as a constraint for messenger scale by using a simple gauge mediation model, taking the following simple messenger sector, where Φ andΦ represent messenger fields. In this set up, the stop mass m 0 is given by In the second line, we used Eq.(2.6) and Eq. (2.8). From this expression we obtain the following formula, (4.3) In order to realize the appropriate Higgs mass, we take m 0 ≃ 10 TeV [20]. Therefore, from Eq.(4.3) M Φ 10 10−11 GeV is required for m a ≃ 10 GeV, λ ′ ≃ 1 to obtain f a 10 12−14 GeV.   Colored region is excluded by BBN or gravitino overproduction. The abundance of the gravitino produced from thermal plasma(R-axion decay) exceeds that of DM in the blue(red) region. Here we have set mg = 10TeV.